Find the Local Maxima and Minima square root of x^4-17x^2+81

$\sqrt{x^{4} - 17 x^{2} + 81}$

Step 1

Write $\sqrt{x^{4} - 17 x^{2} + 81}$ as a function.

$f (x) = \sqrt{x^{4} - 17 x^{2} + 81}$

Step 2

Find the first derivative of the function.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{4} - 17 x^{2} + 81}$ as ${(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}]$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x^{4} - 17 x^{2} + 81$ .

To apply the Chain Rule, set $u$ as $x^{4} - 17 x^{2} + 81$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x^{4} - 17 x^{2} + 81]$

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{2}$ .

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x^{4} - 17 x^{2} + 81]$

Replace all occurrences of $u$ with $x^{4} - 17 x^{2} + 81$ .

$\frac{1}{2} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2} - 1} \frac{d}{d x} [x^{4} - 17 x^{2} + 81]$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x^{4} - 17 x^{2} + 81]$

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{1}{2} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x^{4} - 17 x^{2} + 81]$

Combine the numerators over the common denominator.

$\frac{1}{2} {(x^{4} - 17 x^{2} + 81)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x^{4} - 17 x^{2} + 81]$

Simplify the numerator.

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(x^{4} - 17 x^{2} + 81)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x^{4} - 17 x^{2} + 81]$

Subtract $2$ from $1$ .

$\frac{1}{2} {(x^{4} - 17 x^{2} + 81)}^{\frac{- 1}{2}} \frac{d}{d x} [x^{4} - 17 x^{2} + 81]$

Combine fractions.

Move the negative in front of the fraction.

$\frac{1}{2} {(x^{4} - 17 x^{2} + 81)}^{- \frac{1}{2}} \frac{d}{d x} [x^{4} - 17 x^{2} + 81]$

Combine $\frac{1}{2}$ and ${(x^{4} - 17 x^{2} + 81)}^{- \frac{1}{2}}$ .

$\frac{{(x^{4} - 17 x^{2} + 81)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x^{4} - 17 x^{2} + 81]$

Move ${(x^{4} - 17 x^{2} + 81)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \frac{d}{d x} [x^{4} - 17 x^{2} + 81]$

By the Sum Rule, the derivative of $x^{4} - 17 x^{2} + 81$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 17 x^{2}] + \frac{d}{d x} [81]$ .

$\frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 17 x^{2}] + \frac{d}{d x} [81])$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} (4 x^{3} + \frac{d}{d x} [- 17 x^{2}] + \frac{d}{d x} [81])$

Since $- 17$ is constant with respect to $x$ , the derivative of $- 17 x^{2}$ with respect to $x$ is $- 17 \frac{d}{d x} [x^{2}]$ .

$\frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} (4 x^{3} - 17 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [81])$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} (4 x^{3} - 17 (2 x) + \frac{d}{d x} [81])$

Multiply $2$ by $- 17$ .

$\frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} (4 x^{3} - 34 x + \frac{d}{d x} [81])$

Since $81$ is constant with respect to $x$ , the derivative of $81$ with respect to $x$ is $0$ .

$\frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} (4 x^{3} - 34 x + 0)$

Add $4 x^{3} - 34 x$ and $0$ .

$\frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} (4 x^{3} - 34 x)$

Simplify.

Reorder the factors of $\frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} (4 x^{3} - 34 x)$ .

$(4 x^{3} - 34 x) \frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$

Multiply $4 x^{3} - 34 x$ by $\frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ .

$\frac{4 x^{3} - 34 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$

Factor $2 x$ out of $4 x^{3} - 34 x$ .

Factor $2 x$ out of $4 x^{3}$ .

$\frac{2 x (2 x^{2}) - 34 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$

Factor $2 x$ out of $- 34 x$ .

$\frac{2 x (2 x^{2}) + 2 x (- 17)}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$

Factor $2 x$ out of $2 x (2 x^{2}) + 2 x (- 17)$ .

$\frac{2 x (2 x^{2} - 17)}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$

Cancel the common factor.

$\frac{2 x (2 x^{2} - 17)}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$

Rewrite the expression.

$\frac{x (2 x^{2} - 17)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$

Step 3

Find the second derivative of the function.

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x (2 x^{2} - 17)$ and $g (x) = {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} \frac{d}{d x} (x (2 x^{2} - 17)) - x (2 x^{2} - 17) \frac{d}{d x} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{({(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}})}^{2}}$

Multiply the exponents in ${({(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} \frac{d}{d x} (x (2 x^{2} - 17)) - x (2 x^{2} - 17) \frac{d}{d x} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2} \cdot 2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

Rewrite the expression.

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} \frac{d}{d x} (x (2 x^{2} - 17)) - x (2 x^{2} - 17) \frac{d}{d x} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{x^{4} - 17 x^{2} + 81}$

Simplify.

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} \frac{d}{d x} (x (2 x^{2} - 17)) - x (2 x^{2} - 17) \frac{d}{d x} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{x^{4} - 17 x^{2} + 81}$

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = 2 x^{2} - 17$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (x \frac{d}{d x} (2 x^{2} - 17) + (2 x^{2} - 17) \frac{d}{d x} (x)) - x (2 x^{2} - 17) \frac{d}{d x} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{x^{4} - 17 x^{2} + 81}$

Differentiate.

By the Sum Rule, the derivative of $2 x^{2} - 17$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 17]$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (x (\frac{d}{d x} (2 x^{2}) + \frac{d}{d x} (- 17)) + (2 x^{2} - 17) \frac{d}{d x} (x)) - x (2 x^{2} - 17) \frac{d}{d x} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{x^{4} - 17 x^{2} + 81}$

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{2}$ with respect to $x$ is $2 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (x (2 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 17)) + (2 x^{2} - 17) \frac{d}{d x} (x)) - x (2 x^{2} - 17) \frac{d}{d x} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{x^{4} - 17 x^{2} + 81}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (x (2 (2 x) + \frac{d}{d x} (- 17)) + (2 x^{2} - 17) \frac{d}{d x} (x)) - x (2 x^{2} - 17) \frac{d}{d x} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{x^{4} - 17 x^{2} + 81}$

Multiply $2$ by $2$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (x (4 x + \frac{d}{d x} (- 17)) + (2 x^{2} - 17) \frac{d}{d x} (x)) - x (2 x^{2} - 17) \frac{d}{d x} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{x^{4} - 17 x^{2} + 81}$

Since $- 17$ is constant with respect to $x$ , the derivative of $- 17$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (x (4 x + 0) + (2 x^{2} - 17) \frac{d}{d x} (x)) - x (2 x^{2} - 17) \frac{d}{d x} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{x^{4} - 17 x^{2} + 81}$

Add $4 x$ and $0$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (x (4 x) + (2 x^{2} - 17) \frac{d}{d x} (x)) - x (2 x^{2} - 17) \frac{d}{d x} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{x^{4} - 17 x^{2} + 81}$

Raise $x$ to the power of $1$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (4 (x \cdot x) + (2 x^{2} - 17) \frac{d}{d x} (x)) - x (2 x^{2} - 17) \frac{d}{d x} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{x^{4} - 17 x^{2} + 81}$

Raise $x$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (4 x^{1 + 1} + (2 x^{2} - 17) \frac{d}{d x} (x)) - x (2 x^{2} - 17) \frac{d}{d x} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{x^{4} - 17 x^{2} + 81}$

Differentiate using the Power Rule.

Add $1$ and $1$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (4 x^{2} + (2 x^{2} - 17) \frac{d}{d x} (x)) - x (2 x^{2} - 17) \frac{d}{d x} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{x^{4} - 17 x^{2} + 81}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (4 x^{2} + (2 x^{2} - 17) \cdot 1) - x (2 x^{2} - 17) \frac{d}{d x} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{x^{4} - 17 x^{2} + 81}$

Simplify by adding terms.

Multiply $2 x^{2} - 17$ by $1$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (4 x^{2} + 2 x^{2} - 17) - x (2 x^{2} - 17) \frac{d}{d x} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{x^{4} - 17 x^{2} + 81}$

Add $4 x^{2}$ and $2 x^{2}$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - x (2 x^{2} - 17) \frac{d}{d x} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{x^{4} - 17 x^{2} + 81}$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x^{4} - 17 x^{2} + 81$ .

To apply the Chain Rule, set $u_{1}$ as $x^{4} - 17 x^{2} + 81$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - x (2 x^{2} - 17) (\frac{d}{d u (1)} ({u_{1}}^{\frac{1}{2}}) \frac{d}{d x} (x^{4} - 17 x^{2} + 81))}{x^{4} - 17 x^{2} + 81}$

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{1}{2}$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - x (2 x^{2} - 17) (\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} (x^{4} - 17 x^{2} + 81))}{x^{4} - 17 x^{2} + 81}$

Replace all occurrences of $u_{1}$ with $x^{4} - 17 x^{2} + 81$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - x (2 x^{2} - 17) (\frac{1}{2} \cdot {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2} - 1} \frac{d}{d x} (x^{4} - 17 x^{2} + 81))}{x^{4} - 17 x^{2} + 81}$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - x (2 x^{2} - 17) (\frac{1}{2} \cdot {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (x^{4} - 17 x^{2} + 81))}{x^{4} - 17 x^{2} + 81}$

Combine $- 1$ and $\frac{2}{2}$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - x (2 x^{2} - 17) (\frac{1}{2} \cdot {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (x^{4} - 17 x^{2} + 81))}{x^{4} - 17 x^{2} + 81}$

Combine the numerators over the common denominator.

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - x (2 x^{2} - 17) (\frac{1}{2} \cdot {(x^{4} - 17 x^{2} + 81)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (x^{4} - 17 x^{2} + 81))}{x^{4} - 17 x^{2} + 81}$

Simplify the numerator.

Multiply $- 1$ by $2$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - x (2 x^{2} - 17) (\frac{1}{2} \cdot {(x^{4} - 17 x^{2} + 81)}^{\frac{1 - 2}{2}} \frac{d}{d x} (x^{4} - 17 x^{2} + 81))}{x^{4} - 17 x^{2} + 81}$

Subtract $2$ from $1$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - x (2 x^{2} - 17) (\frac{1}{2} \cdot {(x^{4} - 17 x^{2} + 81)}^{\frac{- 1}{2}} \frac{d}{d x} (x^{4} - 17 x^{2} + 81))}{x^{4} - 17 x^{2} + 81}$

Combine fractions.

Move the negative in front of the fraction.

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - x (2 x^{2} - 17) (\frac{1}{2} \cdot {(x^{4} - 17 x^{2} + 81)}^{- \frac{1}{2}} \frac{d}{d x} (x^{4} - 17 x^{2} + 81))}{x^{4} - 17 x^{2} + 81}$

Combine $\frac{1}{2}$ and ${(x^{4} - 17 x^{2} + 81)}^{- \frac{1}{2}}$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - x (2 x^{2} - 17) (\frac{{(x^{4} - 17 x^{2} + 81)}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (x^{4} - 17 x^{2} + 81))}{x^{4} - 17 x^{2} + 81}$

Move ${(x^{4} - 17 x^{2} + 81)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - x (2 x^{2} - 17) (\frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (x^{4} - 17 x^{2} + 81))}{x^{4} - 17 x^{2} + 81}$

Combine $\frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ and $x$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (2 x^{2} - 17) \frac{d}{d x} (x^{4} - 17 x^{2} + 81)}{x^{4} - 17 x^{2} + 81}$

By the Sum Rule, the derivative of $x^{4} - 17 x^{2} + 81$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 17 x^{2}] + \frac{d}{d x} [81]$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot ((2 x^{2} - 17) (\frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 17 x^{2}) + \frac{d}{d x} (81)))}{x^{4} - 17 x^{2} + 81}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot ((2 x^{2} - 17) (4 x^{3} + \frac{d}{d x} (- 17 x^{2}) + \frac{d}{d x} (81)))}{x^{4} - 17 x^{2} + 81}$

Since $- 17$ is constant with respect to $x$ , the derivative of $- 17 x^{2}$ with respect to $x$ is $- 17 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot ((2 x^{2} - 17) (4 x^{3} - 17 \frac{d}{d x} x^{2} + \frac{d}{d x} (81)))}{x^{4} - 17 x^{2} + 81}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot ((2 x^{2} - 17) (4 x^{3} - 17 (2 x) + \frac{d}{d x} (81)))}{x^{4} - 17 x^{2} + 81}$

Multiply $2$ by $- 17$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot ((2 x^{2} - 17) (4 x^{3} - 34 x + \frac{d}{d x} (81)))}{x^{4} - 17 x^{2} + 81}$

Since $81$ is constant with respect to $x$ , the derivative of $81$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot ((2 x^{2} - 17) (4 x^{3} - 34 x + 0))}{x^{4} - 17 x^{2} + 81}$

Add $4 x^{3} - 34 x$ and $0$ .

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2} - 17) - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot ((2 x^{2} - 17) (4 x^{3} - 34 x))}{x^{4} - 17 x^{2} + 81}$

Simplify.

Simplify the numerator.

Apply the distributive property.

$f'' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (6 x^{2}) + {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} \cdot - 17 - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot ((2 x^{2} - 17) (4 x^{3} - 34 x))}{x^{4} - 17 x^{2} + 81}$

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} + {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} \cdot - 17 - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot ((2 x^{2} - 17) (4 x^{3} - 34 x))}{x^{4} - 17 x^{2} + 81}$

Move $- 17$ to the left of ${(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 \cdot {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot ((2 x^{2} - 17) (4 x^{3} - 34 x))}{x^{4} - 17 x^{2} + 81}$

Apply the distributive property.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + (- \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (2 x^{2}) - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot - 17) (4 x^{3} - 34 x)}{x^{4} - 17 x^{2} + 81}$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ into the numerator.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + (\frac{- x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (2 x^{2}) - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot - 17) (4 x^{3} - 34 x)}{x^{4} - 17 x^{2} + 81}$

Factor $2$ out of $2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + (\frac{- x}{2 ({(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}})} \cdot (2 x^{2}) - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot - 17) (4 x^{3} - 34 x)}{x^{4} - 17 x^{2} + 81}$

Factor $2$ out of $2 x^{2}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + (\frac{- x}{2 ({(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}})} \cdot (2 (x^{2})) - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot - 17) (4 x^{3} - 34 x)}{x^{4} - 17 x^{2} + 81}$

Cancel the common factor.

Rewrite the expression.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + (\frac{- x}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot x^{2} - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot - 17) (4 x^{3} - 34 x)}{x^{4} - 17 x^{2} + 81}$

Combine $\frac{- x}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ and $x^{2}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + (\frac{- x \cdot x^{2}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot - 17) (4 x^{3} - 34 x)}{x^{4} - 17 x^{2} + 81}$

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + (\frac{- (x^{2} x)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot - 17) (4 x^{3} - 34 x)}{x^{4} - 17 x^{2} + 81}$

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + (\frac{- x^{2 + 1}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot - 17) (4 x^{3} - 34 x)}{x^{4} - 17 x^{2} + 81}$

Add $2$ and $1$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + (\frac{- x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot - 17) (4 x^{3} - 34 x)}{x^{4} - 17 x^{2} + 81}$

Multiply $- \frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot - 17$ .

Multiply $- 17$ by $- 1$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + (\frac{- x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + 17 (\frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}})) (4 x^{3} - 34 x)}{x^{4} - 17 x^{2} + 81}$

Combine $17$ and $\frac{x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + (\frac{- x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}) (4 x^{3} - 34 x)}{x^{4} - 17 x^{2} + 81}$

Move the negative in front of the fraction.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + (- \frac{x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}) (4 x^{3} - 34 x)}{x^{4} - 17 x^{2} + 81}$

Expand $(- \frac{x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}) (4 x^{3} - 34 x)$ using the FOIL Method.

Apply the distributive property.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3}) - \frac{x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3} - 34 x)}{x^{4} - 17 x^{2} + 81}$

Apply the distributive property.

Simplify and combine like terms.

Simplify each term.

Multiply $- \frac{x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} (4 x^{3})$ .

Multiply $4$ by $- 1$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - 4 \frac{x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot x^{3} - \frac{x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3}) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Combine $- 4$ and $\frac{x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{- 4 x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot x^{3} - \frac{x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3}) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Combine $\frac{- 4 x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ and $x^{3}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{- 4 x^{3} x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3}) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{- 4 (x^{3} x^{3})}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3}) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{- 4 x^{3 + 3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3}) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Add $3$ and $3$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{- 4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3}) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Move the negative in front of the fraction.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{(4) x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3}) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Multiply $- \frac{x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} (- 34 x)$ .

Multiply $- 34$ by $- 1$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + 34 (\frac{x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}) \cdot x + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3}) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Combine $34$ and $\frac{x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot x + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3}) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Combine $\frac{34 x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ and $x$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{3} x}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3}) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Raise $x$ to the power of $1$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 (x \cdot x^{3})}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3}) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{1 + 3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3}) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Add $1$ and $3$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3}) + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + 4 (\frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}) \cdot x^{3} + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + 2 \cdot (2 (\frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}})) \cdot x^{3} + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Cancel the common factor.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + 2 \cdot (2 (\frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}})) \cdot x^{3} + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Rewrite the expression.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + 2 (\frac{17 x}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}) \cdot x^{3} + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Combine $2$ and $\frac{17 x}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{2 (17 x)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot x^{3} + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Multiply $17$ by $2$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot x^{3} + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Multiply $\frac{34 x}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} x^{3}$ .

Combine $\frac{34 x}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ and $x^{3}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x \cdot x^{3}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Multiply $x$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 (x^{3} x)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Multiply $x^{3}$ by $x$ .

Raise $x$ to the power of $1$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 (x^{3} x)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{3 + 1}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Add $3$ and $1$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (- 34 x)}{x^{4} - 17 x^{2} + 81}$

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - 34 \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot x}{x^{4} - 17 x^{2} + 81}$

Cancel the common factor of $2$ .

Factor $2$ out of $- 34$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + 2 \cdot (- 17 \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}) \cdot x}{x^{4} - 17 x^{2} + 81}$

Cancel the common factor.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + 2 \cdot (- 17 \frac{17 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}) \cdot x}{x^{4} - 17 x^{2} + 81}$

Rewrite the expression.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - 17 \frac{17 x}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot x}{x^{4} - 17 x^{2} + 81}$

Combine $- 17$ and $\frac{17 x}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{- 17 (17 x)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot x}{x^{4} - 17 x^{2} + 81}$

Multiply $17$ by $- 17$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{- 289 x}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot x}{x^{4} - 17 x^{2} + 81}$

Move the negative in front of the fraction.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{(289) x}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot x}{x^{4} - 17 x^{2} + 81}$

Multiply $- \frac{289 x}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} x$ .

Combine $x$ and $\frac{289 x}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{x (289 x)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Raise $x$ to the power of $1$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{289 (x \cdot x)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Raise $x$ to the power of $1$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{289 (x \cdot x)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{289 x^{1 + 1}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Add $1$ and $1$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{289 x^{2}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Add $\frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ and $\frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + 2 (\frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}) - \frac{289 x^{2}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

To write $2 \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + 2 (\frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}) \cdot \frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{289 x^{2}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Combine $2 \frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ and $\frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{4 x^{6}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{2 (\frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}) \cdot {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{289 x^{2}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Combine the numerators over the common denominator.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{- 4 x^{6} + 2 (\frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}) \cdot {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{289 x^{2}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Reorder terms.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{- 4 x^{6} + 2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (\frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}})}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{289 x^{2}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Combine the numerators over the common denominator.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{- 4 x^{6} + 2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (\frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}) - 289 x^{2}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Simplify the numerator.

Cancel the common factor of ${(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}$ .

Factor ${(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}$ out of $2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{- 4 x^{6} + {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} \cdot (2 (\frac{34 x^{4}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}})) - 289 x^{2}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Cancel the common factor.

Rewrite the expression.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{- 4 x^{6} + 2 (34 x^{4}) - 289 x^{2}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Multiply $34$ by $2$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{- 4 x^{6} + 68 x^{4} - 289 x^{2}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Rewrite $- 4 x^{6} + 68 x^{4} - 289 x^{2}$ in a factored form.

Factor $x^{2}$ out of $- 4 x^{6} + 68 x^{4} - 289 x^{2}$ .

Factor $x^{2}$ out of $- 4 x^{6}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} (- 4 x^{4}) + 68 x^{4} - 289 x^{2}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Factor $x^{2}$ out of $68 x^{4}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} (- 4 x^{4}) + x^{2} (68 x^{2}) - 289 x^{2}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Factor $x^{2}$ out of $- 289 x^{2}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} (- 4 x^{4}) + x^{2} (68 x^{2}) + x^{2} \cdot - 289}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Factor $x^{2}$ out of $x^{2} (- 4 x^{4}) + x^{2} (68 x^{2})$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} (- 4 x^{4} + 68 x^{2}) + x^{2} \cdot - 289}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Factor $x^{2}$ out of $x^{2} (- 4 x^{4} + 68 x^{2}) + x^{2} \cdot - 289$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} (- 4 x^{4} + 68 x^{2} - 289)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} (- 4 {(x^{2})}^{2} + 68 x^{2} - 289)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Let $u_{2} = x^{2}$ . Substitute $u_{2}$ for all occurrences of $x^{2}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} (- 4 {u_{2}}^{2} + 68 u_{2} - 289)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 4 \cdot - 289 = 1156$ and whose sum is $b = 68$ .

Factor $68$ out of $68 u_{2}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} (- 4 {u_{2}}^{2} + 68 (u_{2}) - 289)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Rewrite $68$ as $34$ plus $34$

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} (- 4 {u_{2}}^{2} + (34 + 34) u_{2} - 289)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Apply the distributive property.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} (- 4 {u_{2}}^{2} + 34 u_{2} + 34 u_{2} - 289)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} ((- 4 {u_{2}}^{2} + 34 u_{2}) + 34 u_{2} - 289)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Factor out the greatest common factor (GCF) from each group.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} (2 u_{2} (- 2 u_{2} + 17) - 17 (- 2 u_{2} + 17))}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Factor the polynomial by factoring out the greatest common factor, $- 2 u_{2} + 17$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} ((- 2 u_{2} + 17) (2 u_{2} - 17))}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Replace all occurrences of $u_{2}$ with $x^{2}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} (- 2 x^{2} + 17) (2 x^{2} - 17)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Combine exponents.

Factor $- 1$ out of $- 2 x^{2}$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} (- (2 x^{2}) + 17) (2 x^{2} - 17)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Rewrite $17$ as $- 1 (- 17)$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} (- (2 x^{2}) - 1 \cdot - 17) (2 x^{2} - 17)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Factor $- 1$ out of $- (2 x^{2}) - 1 (- 17)$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} (- (2 x^{2} - 17)) (2 x^{2} - 17)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Rewrite $- (2 x^{2} - 17)$ as $- 1 (2 x^{2} - 17)$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} (- 1 (2 x^{2} - 17)) (2 x^{2} - 17)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Raise $2 x^{2} - 17$ to the power of $1$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} \cdot (- 1 ((2 x^{2} - 17) (2 x^{2} - 17)))}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Raise $2 x^{2} - 17$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} \cdot (- 1 {(2 x^{2} - 17)}^{1 + 1})}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Add $1$ and $1$ .

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{x^{2} \cdot (- 1 {(2 x^{2} - 17)}^{2})}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Factor out negative.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} + \frac{- x^{2} {(2 x^{2} - 17)}^{2}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Move the negative in front of the fraction.

$f'' (x) = \frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - \frac{x^{2} {(2 x^{2} - 17)}^{2}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

To write $6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2}$ as a fraction with a common denominator, multiply by $\frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{\frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - \frac{x^{2} {(2 x^{2} - 17)}^{2}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{x^{4} - 17 x^{2} + 81}$

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - x^{2} {(2 x^{2} - 17)}^{2}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{x^{4} - 17 x^{2} + 81}$

To write $- 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ .

Combine $- 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}$ and $\frac{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ .

$f'' (x) = \frac{\frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - x^{2} {(2 x^{2} - 17)}^{2}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} + \frac{- 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - x^{2} {(2 x^{2} - 17)}^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Rewrite $\frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} x^{2} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} - x^{2} {(2 x^{2} - 17)}^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ in a factored form.

Multiply ${(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}$ by ${(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}$ by adding the exponents.

Move ${(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{\frac{6 ({(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}) x^{2} - x^{2} {(2 x^{2} - 17)}^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{\frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2} + \frac{1}{2}} x^{2} - x^{2} {(2 x^{2} - 17)}^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{1 + 1}{2}} x^{2} - x^{2} {(2 x^{2} - 17)}^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Add $1$ and $1$ .

$f'' (x) = \frac{\frac{6 {(x^{4} - 17 x^{2} + 81)}^{\frac{2}{2}} x^{2} - x^{2} {(2 x^{2} - 17)}^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Divide $2$ by $2$ .

$f'' (x) = \frac{\frac{6 (x^{4} - 17 x^{2} + 81) x^{2} - x^{2} {(2 x^{2} - 17)}^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Simplify $6 {(x^{4} - 17 x^{2} + 81)}^{1} x^{2}$ .

Apply the distributive property.

$f'' (x) = \frac{\frac{(6 x^{4} + 6 (- 17 x^{2}) + 6 \cdot 81) x^{2} - x^{2} {(2 x^{2} - 17)}^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Simplify.

Multiply $- 17$ by $6$ .

$f'' (x) = \frac{\frac{(6 x^{4} - 102 x^{2} + 6 \cdot 81) x^{2} - x^{2} {(2 x^{2} - 17)}^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Multiply $6$ by $81$ .

$f'' (x) = \frac{\frac{(6 x^{4} - 102 x^{2} + 486) x^{2} - x^{2} {(2 x^{2} - 17)}^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Apply the distributive property.

$f'' (x) = \frac{\frac{6 x^{4} x^{2} - 102 x^{2} x^{2} + 486 x^{2} - x^{2} {(2 x^{2} - 17)}^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Simplify.

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$f'' (x) = \frac{\frac{6 (x^{2} x^{4}) - 102 x^{2} x^{2} + 486 x^{2} - x^{2} {(2 x^{2} - 17)}^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{\frac{6 x^{2 + 4} - 102 x^{2} x^{2} + 486 x^{2} - x^{2} {(2 x^{2} - 17)}^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Add $2$ and $4$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{2} x^{2} + 486 x^{2} - x^{2} {(2 x^{2} - 17)}^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 (x^{2} x^{2}) + 486 x^{2} - x^{2} {(2 x^{2} - 17)}^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{2 + 2} + 486 x^{2} - x^{2} {(2 x^{2} - 17)}^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Add $2$ and $2$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - x^{2} {(2 x^{2} - 17)}^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Rewrite ${(2 x^{2} - 17)}^{2}$ as $(2 x^{2} - 17) (2 x^{2} - 17)$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - x^{2} ((2 x^{2} - 17) (2 x^{2} - 17)) - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Expand $(2 x^{2} - 17) (2 x^{2} - 17)$ using the FOIL Method.

Apply the distributive property.

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - x^{2} (2 x^{2} (2 x^{2} - 17) - 17 (2 x^{2} - 17)) - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Apply the distributive property.

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - x^{2} (2 x^{2} (2 x^{2}) + 2 x^{2} \cdot - 17 - 17 (2 x^{2} - 17)) - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Apply the distributive property.

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - x^{2} (2 x^{2} (2 x^{2}) + 2 x^{2} \cdot - 17 - 17 (2 x^{2}) - 17 \cdot - 17) - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - x^{2} (2 \cdot (2 x^{2} x^{2}) + 2 x^{2} \cdot - 17 - 17 (2 x^{2}) - 17 \cdot - 17) - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - x^{2} (2 \cdot (2 (x^{2} x^{2})) + 2 x^{2} \cdot - 17 - 17 (2 x^{2}) - 17 \cdot - 17) - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - x^{2} (2 \cdot (2 x^{2 + 2}) + 2 x^{2} \cdot - 17 - 17 (2 x^{2}) - 17 \cdot - 17) - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Add $2$ and $2$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - x^{2} (2 \cdot (2 x^{4}) + 2 x^{2} \cdot - 17 - 17 (2 x^{2}) - 17 \cdot - 17) - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Multiply $2$ by $2$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - x^{2} (4 x^{4} + 2 x^{2} \cdot - 17 - 17 (2 x^{2}) - 17 \cdot - 17) - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Multiply $- 17$ by $2$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - x^{2} (4 x^{4} - 34 x^{2} - 17 (2 x^{2}) - 17 \cdot - 17) - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Multiply $2$ by $- 17$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - x^{2} (4 x^{4} - 34 x^{2} - 34 x^{2} - 17 \cdot - 17) - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Multiply $- 17$ by $- 17$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - x^{2} (4 x^{4} - 34 x^{2} - 34 x^{2} + 289) - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Subtract $34 x^{2}$ from $- 34 x^{2}$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - x^{2} (4 x^{4} - 68 x^{2} + 289) - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Apply the distributive property.

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - x^{2} (4 x^{4}) - x^{2} (- 68 x^{2}) - x^{2} \cdot 289 - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Simplify.

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 1 \cdot (4 x^{2} x^{4}) - x^{2} (- 68 x^{2}) - x^{2} \cdot 289 - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 1 \cdot (4 x^{2} x^{4}) - 1 \cdot (- 68 x^{2} x^{2}) - x^{2} \cdot 289 - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Multiply $289$ by $- 1$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 1 \cdot (4 x^{2} x^{4}) - 1 \cdot (- 68 x^{2} x^{2}) - 289 x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Simplify each term.

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 1 \cdot (4 (x^{4} x^{2})) - 1 \cdot (- 68 x^{2} x^{2}) - 289 x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 1 \cdot (4 x^{4 + 2}) - 1 \cdot (- 68 x^{2} x^{2}) - 289 x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Add $4$ and $2$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 1 \cdot (4 x^{6}) - 1 \cdot (- 68 x^{2} x^{2}) - 289 x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Multiply $- 1$ by $4$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 4 x^{6} - 1 \cdot (- 68 x^{2} x^{2}) - 289 x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 4 x^{6} - 1 \cdot (- 68 (x^{2} x^{2})) - 289 x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 4 x^{6} - 1 \cdot (- 68 x^{2 + 2}) - 289 x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Add $2$ and $2$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 4 x^{6} - 1 \cdot (- 68 x^{4}) - 289 x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Multiply $- 1$ by $- 68$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 4 x^{6} + 68 x^{4} - 289 x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Multiply ${(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}$ by ${(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}$ by adding the exponents.

Move ${(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 4 x^{6} + 68 x^{4} - 289 x^{2} - 17 ({(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}})}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 4 x^{6} + 68 x^{4} - 289 x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2} + \frac{1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 4 x^{6} + 68 x^{4} - 289 x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{1 + 1}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Add $1$ and $1$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 4 x^{6} + 68 x^{4} - 289 x^{2} - 17 {(x^{4} - 17 x^{2} + 81)}^{\frac{2}{2}}}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Divide $2$ by $2$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 4 x^{6} + 68 x^{4} - 289 x^{2} - 17 (x^{4} - 17 x^{2} + 81)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Simplify $- 17 {(x^{4} - 17 x^{2} + 81)}^{1}$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 4 x^{6} + 68 x^{4} - 289 x^{2} - 17 (x^{4} - 17 x^{2} + 81)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Apply the distributive property.

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 4 x^{6} + 68 x^{4} - 289 x^{2} - 17 x^{4} - 17 (- 17 x^{2}) - 17 \cdot 81}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Simplify.

Multiply $- 17$ by $- 17$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 4 x^{6} + 68 x^{4} - 289 x^{2} - 17 x^{4} + 289 x^{2} - 17 \cdot 81}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Multiply $- 17$ by $81$ .

$f'' (x) = \frac{\frac{6 x^{6} - 102 x^{4} + 486 x^{2} - 4 x^{6} + 68 x^{4} - 289 x^{2} - 17 x^{4} + 289 x^{2} - 1377}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Subtract $4 x^{6}$ from $6 x^{6}$ .

$f'' (x) = \frac{\frac{2 x^{6} - 102 x^{4} + 486 x^{2} + 68 x^{4} - 289 x^{2} - 17 x^{4} + 289 x^{2} - 1377}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Add $- 102 x^{4}$ and $68 x^{4}$ .

$f'' (x) = \frac{\frac{2 x^{6} - 34 x^{4} + 486 x^{2} - 289 x^{2} - 17 x^{4} + 289 x^{2} - 1377}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Subtract $17 x^{4}$ from $- 34 x^{4}$ .

$f'' (x) = \frac{\frac{2 x^{6} - 51 x^{4} + 486 x^{2} - 289 x^{2} + 289 x^{2} - 1377}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Subtract $289 x^{2}$ from $486 x^{2}$ .

$f'' (x) = \frac{\frac{2 x^{6} - 51 x^{4} + 197 x^{2} + 289 x^{2} - 1377}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Add $197 x^{2}$ and $289 x^{2}$ .

$f'' (x) = \frac{\frac{2 x^{6} - 51 x^{4} + 486 x^{2} - 1377}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$

Combine terms.

Rewrite $\frac{\frac{2 x^{6} - 51 x^{4} + 486 x^{2} - 1377}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}}{x^{4} - 17 x^{2} + 81}$ as a product.

$f'' (x) = \frac{2 x^{6} - 51 x^{4} + 486 x^{2} - 1377}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot \frac{1}{x^{4} - 17 x^{2} + 81}$

Multiply $\frac{2 x^{6} - 51 x^{4} + 486 x^{2} - 1377}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ by $\frac{1}{x^{4} - 17 x^{2} + 81}$ .

$f'' (x) = \frac{2 x^{6} - 51 x^{4} + 486 x^{2} - 1377}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (x^{4} - 17 x^{2} + 81)}$

Multiply ${(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}$ by $x^{4} - 17 x^{2} + 81$ by adding the exponents.

Multiply ${(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}$ by $x^{4} - 17 x^{2} + 81$ .

Raise $x^{4} - 17 x^{2} + 81$ to the power of $1$ .

$f'' (x) = \frac{2 x^{6} - 51 x^{4} + 486 x^{2} - 1377}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}} (x^{4} - 17 x^{2} + 81)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{2 x^{6} - 51 x^{4} + 486 x^{2} - 1377}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2} + 1}}$

Write $1$ as a fraction with a common denominator.

$f'' (x) = \frac{2 x^{6} - 51 x^{4} + 486 x^{2} - 1377}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2} + \frac{2}{2}}}$

Combine the numerators over the common denominator.

$f'' (x) = \frac{2 x^{6} - 51 x^{4} + 486 x^{2} - 1377}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1 + 2}{2}}}$

Add $1$ and $2$ .

$f'' (x) = \frac{2 x^{6} - 51 x^{4} + 486 x^{2} - 1377}{{(x^{4} - 17 x^{2} + 81)}^{\frac{3}{2}}}$

Step 4

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{x (2 x^{2} - 17)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} = 0$

Step 5

Find the first derivative.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{4} - 17 x^{2} + 81}$ as ${(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}$ .

$f' (x) = \frac{d}{d x} ({(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}})$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x^{4} - 17 x^{2} + 81$ .

To apply the Chain Rule, set $u$ as $x^{4} - 17 x^{2} + 81$ .

$f' (x) = \frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d x} (x^{4} - 17 x^{2} + 81)$

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{2}$ .

$f' (x) = \frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} (x^{4} - 17 x^{2} + 81)$

Replace all occurrences of $u$ with $x^{4} - 17 x^{2} + 81$ .

$f' (x) = \frac{1}{2} \cdot {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2} - 1} \frac{d}{d x} (x^{4} - 17 x^{2} + 81)$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (x) = \frac{1}{2} \cdot {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (x^{4} - 17 x^{2} + 81)$

Combine $- 1$ and $\frac{2}{2}$ .

$f' (x) = \frac{1}{2} \cdot {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (x^{4} - 17 x^{2} + 81)$

Combine the numerators over the common denominator.

$f' (x) = \frac{1}{2} \cdot {(x^{4} - 17 x^{2} + 81)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (x^{4} - 17 x^{2} + 81)$

Simplify the numerator.

Multiply $- 1$ by $2$ .

$f' (x) = \frac{1}{2} \cdot {(x^{4} - 17 x^{2} + 81)}^{\frac{1 - 2}{2}} \frac{d}{d x} (x^{4} - 17 x^{2} + 81)$

Subtract $2$ from $1$ .

$f' (x) = \frac{1}{2} \cdot {(x^{4} - 17 x^{2} + 81)}^{\frac{- 1}{2}} \frac{d}{d x} (x^{4} - 17 x^{2} + 81)$

Combine fractions.

Move the negative in front of the fraction.

$f' (x) = \frac{1}{2} \cdot {(x^{4} - 17 x^{2} + 81)}^{- \frac{1}{2}} \frac{d}{d x} (x^{4} - 17 x^{2} + 81)$

Combine $\frac{1}{2}$ and ${(x^{4} - 17 x^{2} + 81)}^{- \frac{1}{2}}$ .

$f' (x) = \frac{{(x^{4} - 17 x^{2} + 81)}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (x^{4} - 17 x^{2} + 81)$

Move ${(x^{4} - 17 x^{2} + 81)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = \frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (x^{4} - 17 x^{2} + 81)$

By the Sum Rule, the derivative of $x^{4} - 17 x^{2} + 81$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 17 x^{2}] + \frac{d}{d x} [81]$ .

$f' (x) = \frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 17 x^{2}) + \frac{d}{d x} (81))$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f' (x) = \frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3} + \frac{d}{d x} (- 17 x^{2}) + \frac{d}{d x} (81))$

Since $- 17$ is constant with respect to $x$ , the derivative of $- 17 x^{2}$ with respect to $x$ is $- 17 \frac{d}{d x} [x^{2}]$ .

$f' (x) = \frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3} - 17 \frac{d}{d x} x^{2} + \frac{d}{d x} (81))$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f' (x) = \frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3} - 17 (2 x) + \frac{d}{d x} (81))$

Multiply $2$ by $- 17$ .

$f' (x) = \frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3} - 34 x + \frac{d}{d x} (81))$

Since $81$ is constant with respect to $x$ , the derivative of $81$ with respect to $x$ is $0$ .

$f' (x) = \frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3} - 34 x + 0)$

Add $4 x^{3} - 34 x$ and $0$ .

$f' (x) = \frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} \cdot (4 x^{3} - 34 x)$

Simplify.

Reorder the factors of $\frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} (4 x^{3} - 34 x)$ .

$f' (x) = (4 x^{3} - 34 x) (\frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}})$

Multiply $4 x^{3} - 34 x$ by $\frac{1}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ .

$f' (x) = \frac{4 x^{3} - 34 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$

Factor $2 x$ out of $4 x^{3} - 34 x$ .

Factor $2 x$ out of $4 x^{3}$ .

$f' (x) = \frac{2 x (2 x^{2}) - 34 x}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$

Factor $2 x$ out of $- 34 x$ .

$f' (x) = \frac{2 x (2 x^{2}) + 2 x (- 17)}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$

Factor $2 x$ out of $2 x (2 x^{2}) + 2 x (- 17)$ .

$f' (x) = \frac{2 x (2 x^{2} - 17)}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$

Cancel the common factor.

$f' (x) = \frac{2 x (2 x^{2} - 17)}{2 {(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$

Rewrite the expression.

$f' (x) = \frac{x (2 x^{2} - 17)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$

The first derivative of $f (x)$ with respect to $x$ is $\frac{x (2 x^{2} - 17)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$ .

$\frac{x (2 x^{2} - 17)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $\frac{x (2 x^{2} - 17)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} = 0$ .

Set the first derivative equal to $0$ .

$\frac{x (2 x^{2} - 17)}{{(x^{4} - 17 x^{2} + 81)}^{\frac{1}{2}}} = 0$

Set the numerator equal to zero.

$x (2 x^{2} - 17) = 0$

Solve the equation for $x$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$2 x^{2} - 17 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $2 x^{2} - 17$ equal to $0$ and solve for $x$ .

Set $2 x^{2} - 17$ equal to $0$ .

$2 x^{2} - 17 = 0$

Solve $2 x^{2} - 17 = 0$ for $x$ .

Add $17$ to both sides of the equation.

$2 x^{2} = 17$

Divide each term in $2 x^{2} = 17$ by $2$ and simplify.

Divide each term in $2 x^{2} = 17$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{17}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{17}{2}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{17}{2}$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{17}{2}}$

Simplify $\pm \sqrt{\frac{17}{2}}$ .

Rewrite $\sqrt{\frac{17}{2}}$ as $\frac{\sqrt{17}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{17}}{\sqrt{2}}$

Multiply $\frac{\sqrt{17}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{17}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{17}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{17} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{17} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{17} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{17} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{17} \sqrt{2}}{{\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{17} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{17} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{17} \sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{\sqrt{17} \sqrt{2}}{2^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm \frac{\sqrt{17} \sqrt{2}}{2^{1}}$

Evaluate the exponent.

$x = \pm \frac{\sqrt{17} \sqrt{2}}{2}$

Simplify the numerator.

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{17 \cdot 2}}{2}$

Multiply $17$ by $2$ .

$x = \pm \frac{\sqrt{34}}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{34}}{2}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{34}}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{34}}{2}, - \frac{\sqrt{34}}{2}$

The final solution is all the values that make $x (2 x^{2} - 17) = 0$ true.

$x = 0, \frac{\sqrt{34}}{2}, - \frac{\sqrt{34}}{2}$

Step 7

Find the values where the derivative is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 8

Critical points to evaluate.

$x = 0, \frac{\sqrt{34}}{2}, - \frac{\sqrt{34}}{2}$

Step 9

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{2 {(0)}^{6} - 51 {(0)}^{4} + 486 {(0)}^{2} - 1377}{{({(0)}^{4} - 17 {(0)}^{2} + 81)}^{\frac{3}{2}}}$

Step 10

Evaluate the second derivative.

Simplify the numerator.

Raising $0$ to any positive power yields $0$ .

$\frac{2 \cdot 0 - 51 \cdot 0^{4} + 486 \cdot 0^{2} - 1377}{{(0^{4} - 17 \cdot 0^{2} + 81)}^{\frac{3}{2}}}$

Multiply $2$ by $0$ .

$\frac{0 - 51 \cdot 0^{4} + 486 \cdot 0^{2} - 1377}{{(0^{4} - 17 \cdot 0^{2} + 81)}^{\frac{3}{2}}}$

Raising $0$ to any positive power yields $0$ .

$\frac{0 - 51 \cdot 0 + 486 \cdot 0^{2} - 1377}{{(0^{4} - 17 \cdot 0^{2} + 81)}^{\frac{3}{2}}}$

Multiply $- 51$ by $0$ .

$\frac{0 + 0 + 486 \cdot 0^{2} - 1377}{{(0^{4} - 17 \cdot 0^{2} + 81)}^{\frac{3}{2}}}$

Raising $0$ to any positive power yields $0$ .

$\frac{0 + 0 + 486 \cdot 0 - 1377}{{(0^{4} - 17 \cdot 0^{2} + 81)}^{\frac{3}{2}}}$

Multiply $486$ by $0$ .

$\frac{0 + 0 + 0 - 1377}{{(0^{4} - 17 \cdot 0^{2} + 81)}^{\frac{3}{2}}}$

Add $0$ and $0$ .

$\frac{0 + 0 - 1377}{{(0^{4} - 17 \cdot 0^{2} + 81)}^{\frac{3}{2}}}$

Add $0$ and $0$ .

$\frac{0 - 1377}{{(0^{4} - 17 \cdot 0^{2} + 81)}^{\frac{3}{2}}}$

Subtract $1377$ from $0$ .

$\frac{- 1377}{{(0^{4} - 17 \cdot 0^{2} + 81)}^{\frac{3}{2}}}$

Simplify the denominator.

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$\frac{- 1377}{{(0 - 17 \cdot 0^{2} + 81)}^{\frac{3}{2}}}$

Raising $0$ to any positive power yields $0$ .

$\frac{- 1377}{{(0 - 17 \cdot 0 + 81)}^{\frac{3}{2}}}$

Multiply $- 17$ by $0$ .

$\frac{- 1377}{{(0 + 0 + 81)}^{\frac{3}{2}}}$

Add $0$ and $0$ .

$\frac{- 1377}{{(0 + 81)}^{\frac{3}{2}}}$

Add $0$ and $81$ .

$\frac{- 1377}{81^{\frac{3}{2}}}$

Rewrite $81$ as $9^{2}$ .

$\frac{- 1377}{{(9^{2})}^{\frac{3}{2}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{- 1377}{9^{2 (\frac{3}{2})}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{- 1377}{9^{2 (\frac{3}{2})}}$

Rewrite the expression.

$\frac{- 1377}{9^{3}}$

Raise $9$ to the power of $3$ .

$\frac{- 1377}{729}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 1377$ and $729$ .

Factor $81$ out of $- 1377$ .

$\frac{81 (- 17)}{729}$

Cancel the common factors.

Factor $81$ out of $729$ .

$\frac{81 \cdot - 17}{81 \cdot 9}$

Cancel the common factor.

$\frac{81 \cdot - 17}{81 \cdot 9}$

Rewrite the expression.

$\frac{- 17}{9}$

Move the negative in front of the fraction.

$- \frac{17}{9}$

Step 11

$x = 0$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 0$ is a local maximum

Step 12

Find the y-value when $x = 0$ .

Replace the variable $x$ with $0$ in the expression.

$f (0) = \sqrt{{(0)}^{4} - 17 {(0)}^{2} + 81}$

Simplify the result.

Raising $0$ to any positive power yields $0$ .

$f (0) = \sqrt{0 - 17 {(0)}^{2} + 81}$

Raising $0$ to any positive power yields $0$ .

$f (0) = \sqrt{0 - 17 \cdot 0 + 81}$

Multiply $- 17$ by $0$ .

$f (0) = \sqrt{0 + 0 + 81}$

Add $0$ and $0$ .

$f (0) = \sqrt{0 + 81}$

Add $0$ and $81$ .

$f (0) = \sqrt{81}$

Rewrite $81$ as $9^{2}$ .

$f (0) = \sqrt{9^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$f (0) = 9$

The final answer is $9$ .

$y = 9$

Step 13

Evaluate the second derivative at $x = \frac{\sqrt{34}}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{2 {(\frac{\sqrt{34}}{2})}^{6} - 51 {(\frac{\sqrt{34}}{2})}^{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Step 14

Evaluate the second derivative.

Simplify the numerator.

Apply the product rule to $\frac{\sqrt{34}}{2}$ .

$\frac{2 \frac{{\sqrt{34}}^{6}}{2^{6}} - 51 {(\frac{\sqrt{34}}{2})}^{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Simplify the numerator.

Rewrite ${\sqrt{34}}^{6}$ as $34^{3}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\frac{2 \frac{{(34^{\frac{1}{2}})}^{6}}{2^{6}} - 51 {(\frac{\sqrt{34}}{2})}^{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 \frac{34^{\frac{1}{2} \cdot 6}}{2^{6}} - 51 {(\frac{\sqrt{34}}{2})}^{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine $\frac{1}{2}$ and $6$ .

$\frac{2 \frac{34^{\frac{6}{2}}}{2^{6}} - 51 {(\frac{\sqrt{34}}{2})}^{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6$ .

$\frac{2 \frac{34^{\frac{2 \cdot 3}{2}}}{2^{6}} - 51 {(\frac{\sqrt{34}}{2})}^{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factors.

Factor $2$ out of $2$ .

$\frac{2 \frac{34^{\frac{2 \cdot 3}{2 (1)}}}{2^{6}} - 51 {(\frac{\sqrt{34}}{2})}^{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

$\frac{2 \frac{34^{\frac{2 \cdot 3}{2 \cdot 1}}}{2^{6}} - 51 {(\frac{\sqrt{34}}{2})}^{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{2 \frac{34^{\frac{3}{1}}}{2^{6}} - 51 {(\frac{\sqrt{34}}{2})}^{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Divide $3$ by $1$ .

$\frac{2 \frac{34^{3}}{2^{6}} - 51 {(\frac{\sqrt{34}}{2})}^{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Raise $34$ to the power of $3$ .

$\frac{2 \frac{39304}{2^{6}} - 51 {(\frac{\sqrt{34}}{2})}^{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Raise $2$ to the power of $6$ .

$\frac{2 (\frac{39304}{64}) - 51 {(\frac{\sqrt{34}}{2})}^{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $2$ .

Factor $2$ out of $64$ .

$\frac{2 \frac{39304}{2 (32)} - 51 {(\frac{\sqrt{34}}{2})}^{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

$\frac{2 \frac{39304}{2 \cdot 32} - 51 {(\frac{\sqrt{34}}{2})}^{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{39304}{32} - 51 {(\frac{\sqrt{34}}{2})}^{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $39304$ and $32$ .

Factor $8$ out of $39304$ .

$\frac{\frac{8 (4913)}{32} - 51 {(\frac{\sqrt{34}}{2})}^{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factors.

Factor $8$ out of $32$ .

$\frac{\frac{8 \cdot 4913}{8 \cdot 4} - 51 {(\frac{\sqrt{34}}{2})}^{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

$\frac{\frac{8 \cdot 4913}{8 \cdot 4} - 51 {(\frac{\sqrt{34}}{2})}^{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{4913}{4} - 51 {(\frac{\sqrt{34}}{2})}^{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Apply the product rule to $\frac{\sqrt{34}}{2}$ .

$\frac{\frac{4913}{4} - 51 \frac{{\sqrt{34}}^{4}}{2^{4}} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Simplify the numerator.

Rewrite ${\sqrt{34}}^{4}$ as $34^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\frac{\frac{4913}{4} - 51 \frac{{(34^{\frac{1}{2}})}^{4}}{2^{4}} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{4913}{4} - 51 \frac{34^{\frac{1}{2} \cdot 4}}{2^{4}} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine $\frac{1}{2}$ and $4$ .

$\frac{\frac{4913}{4} - 51 \frac{34^{\frac{4}{2}}}{2^{4}} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

$\frac{\frac{4913}{4} - 51 \frac{34^{\frac{2 \cdot 2}{2}}}{2^{4}} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factors.

Factor $2$ out of $2$ .

$\frac{\frac{4913}{4} - 51 \frac{34^{\frac{2 \cdot 2}{2 (1)}}}{2^{4}} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

$\frac{\frac{4913}{4} - 51 \frac{34^{\frac{2 \cdot 2}{2 \cdot 1}}}{2^{4}} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{4913}{4} - 51 \frac{34^{\frac{2}{1}}}{2^{4}} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Divide $2$ by $1$ .

$\frac{\frac{4913}{4} - 51 \frac{34^{2}}{2^{4}} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Raise $34$ to the power of $2$ .

$\frac{\frac{4913}{4} - 51 \frac{1156}{2^{4}} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Raise $2$ to the power of $4$ .

$\frac{\frac{4913}{4} - 51 (\frac{1156}{16}) + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $1156$ and $16$ .

Factor $4$ out of $1156$ .

$\frac{\frac{4913}{4} - 51 \frac{4 (289)}{16} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factors.

Factor $4$ out of $16$ .

$\frac{\frac{4913}{4} - 51 \frac{4 \cdot 289}{4 \cdot 4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

$\frac{\frac{4913}{4} - 51 \frac{4 \cdot 289}{4 \cdot 4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{4913}{4} - 51 (\frac{289}{4}) + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Multiply $- 51 (\frac{289}{4})$ .

Combine $- 51$ and $\frac{289}{4}$ .

$\frac{\frac{4913}{4} + \frac{- 51 \cdot 289}{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Multiply $- 51$ by $289$ .

$\frac{\frac{4913}{4} + \frac{- 14739}{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Move the negative in front of the fraction.

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 {(\frac{\sqrt{34}}{2})}^{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Apply the product rule to $\frac{\sqrt{34}}{2}$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 \frac{{\sqrt{34}}^{2}}{2^{2}} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite ${\sqrt{34}}^{2}$ as $34$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 \frac{{(34^{\frac{1}{2}})}^{2}}{2^{2}} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 \frac{34^{\frac{1}{2} \cdot 2}}{2^{2}} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 \frac{34^{\frac{2}{2}}}{2^{2}} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 \frac{34^{\frac{2}{2}}}{2^{2}} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 \frac{34^{1}}{2^{2}} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Evaluate the exponent.

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 \frac{34}{2^{2}} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Raise $2$ to the power of $2$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 (\frac{34}{4}) - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $2$ .

Factor $2$ out of $486$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + 2 (243) \frac{34}{4} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Factor $2$ out of $4$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + 2 \cdot 243 \frac{34}{2 \cdot 2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

$\frac{\frac{4913}{4} - \frac{14739}{4} + 2 \cdot 243 \frac{34}{2 \cdot 2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{4913}{4} - \frac{14739}{4} + 243 (\frac{34}{2}) - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine $243$ and $\frac{34}{2}$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + \frac{243 \cdot 34}{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Multiply $243$ by $34$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + \frac{8262}{2} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Divide $8262$ by $2$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + 4131 - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine the numerators over the common denominator.

$\frac{\frac{4913 - 14739}{4} + 4131 - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Subtract $14739$ from $4913$ .

$\frac{\frac{- 9826}{4} + 4131 - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

To write $4131$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\frac{- 9826}{4} + 4131 \cdot \frac{4}{4} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine $4131$ and $\frac{4}{4}$ .

$\frac{\frac{- 9826}{4} + \frac{4131 \cdot 4}{4} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine the numerators over the common denominator.

$\frac{\frac{- 9826 + 4131 \cdot 4}{4} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Simplify the numerator.

Multiply $4131$ by $4$ .

$\frac{\frac{- 9826 + 16524}{4} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Add $- 9826$ and $16524$ .

$\frac{\frac{6698}{4} - 1377}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

To write $- 1377$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\frac{6698}{4} - 1377 \cdot \frac{4}{4}}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine $- 1377$ and $\frac{4}{4}$ .

$\frac{\frac{6698}{4} + \frac{- 1377 \cdot 4}{4}}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine the numerators over the common denominator.

$\frac{\frac{6698 - 1377 \cdot 4}{4}}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Simplify the numerator.

Multiply $- 1377$ by $4$ .

$\frac{\frac{6698 - 5508}{4}}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Subtract $5508$ from $6698$ .

$\frac{\frac{1190}{4}}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $1190$ and $4$ .

Factor $2$ out of $1190$ .

$\frac{\frac{2 (595)}{4}}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{\frac{2 \cdot 595}{2 \cdot 2}}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

$\frac{\frac{2 \cdot 595}{2 \cdot 2}}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{595}{2}}{{({(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Simplify the denominator.

Simplify each term.

Apply the product rule to $\frac{\sqrt{34}}{2}$ .

$\frac{\frac{595}{2}}{{(\frac{{\sqrt{34}}^{4}}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Simplify the numerator.

Rewrite ${\sqrt{34}}^{4}$ as $34^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\frac{\frac{595}{2}}{{(\frac{{(34^{\frac{1}{2}})}^{4}}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{595}{2}}{{(\frac{34^{\frac{1}{2} \cdot 4}}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine $\frac{1}{2}$ and $4$ .

$\frac{\frac{595}{2}}{{(\frac{34^{\frac{4}{2}}}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

$\frac{\frac{595}{2}}{{(\frac{34^{\frac{2 \cdot 2}{2}}}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factors.

Factor $2$ out of $2$ .

$\frac{\frac{595}{2}}{{(\frac{34^{\frac{2 \cdot 2}{2 (1)}}}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

$\frac{\frac{595}{2}}{{(\frac{34^{\frac{2 \cdot 2}{2 \cdot 1}}}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{595}{2}}{{(\frac{34^{\frac{2}{1}}}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Divide $2$ by $1$ .

$\frac{\frac{595}{2}}{{(\frac{34^{2}}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Raise $34$ to the power of $2$ .

$\frac{\frac{595}{2}}{{(\frac{1156}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Raise $2$ to the power of $4$ .

$\frac{\frac{595}{2}}{{(\frac{1156}{16} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $1156$ and $16$ .

Factor $4$ out of $1156$ .

$\frac{\frac{595}{2}}{{(\frac{4 (289)}{16} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factors.

Factor $4$ out of $16$ .

$\frac{\frac{595}{2}}{{(\frac{4 \cdot 289}{4 \cdot 4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

$\frac{\frac{595}{2}}{{(\frac{4 \cdot 289}{4 \cdot 4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Apply the product rule to $\frac{\sqrt{34}}{2}$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{{\sqrt{34}}^{2}}{2^{2}} + 81)}^{\frac{3}{2}}}$

Rewrite ${\sqrt{34}}^{2}$ as $34$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{{(34^{\frac{1}{2}})}^{2}}{2^{2}} + 81)}^{\frac{3}{2}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{34^{\frac{1}{2} \cdot 2}}{2^{2}} + 81)}^{\frac{3}{2}}}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{34^{\frac{2}{2}}}{2^{2}} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{34^{\frac{2}{2}}}{2^{2}} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{34^{1}}{2^{2}} + 81)}^{\frac{3}{2}}}$

Evaluate the exponent.

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{34}{2^{2}} + 81)}^{\frac{3}{2}}}$

Raise $2$ to the power of $2$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 (\frac{34}{4}) + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $34$ and $4$ .

Factor $2$ out of $34$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{2 (17)}{4} + 81)}^{\frac{3}{2}}}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{2 \cdot 17}{2 \cdot 2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{2 \cdot 17}{2 \cdot 2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 (\frac{17}{2}) + 81)}^{\frac{3}{2}}}$

Multiply $- 17 (\frac{17}{2})$ .

Combine $- 17$ and $\frac{17}{2}$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} + \frac{- 17 \cdot 17}{2} + 81)}^{\frac{3}{2}}}$

Multiply $- 17$ by $17$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} + \frac{- 289}{2} + 81)}^{\frac{3}{2}}}$

Move the negative in front of the fraction.

$\frac{\frac{595}{2}}{{(\frac{289}{4} - \frac{289}{2} + 81)}^{\frac{3}{2}}}$

Find the common denominator.

Multiply $\frac{289}{2}$ by $\frac{2}{2}$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - (\frac{289}{2} \cdot \frac{2}{2}) + 81)}^{\frac{3}{2}}}$

Multiply $\frac{289}{2}$ by $\frac{2}{2}$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - \frac{289 \cdot 2}{2 \cdot 2} + 81)}^{\frac{3}{2}}}$

Write $81$ as a fraction with denominator $1$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - \frac{289 \cdot 2}{2 \cdot 2} + \frac{81}{1})}^{\frac{3}{2}}}$

Multiply $\frac{81}{1}$ by $\frac{4}{4}$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - \frac{289 \cdot 2}{2 \cdot 2} + \frac{81}{1} \cdot \frac{4}{4})}^{\frac{3}{2}}}$

Multiply $\frac{81}{1}$ by $\frac{4}{4}$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - \frac{289 \cdot 2}{2 \cdot 2} + \frac{81 \cdot 4}{4})}^{\frac{3}{2}}}$

Multiply $2$ by $2$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - \frac{289 \cdot 2}{4} + \frac{81 \cdot 4}{4})}^{\frac{3}{2}}}$

Combine the numerators over the common denominator.

$\frac{\frac{595}{2}}{{(\frac{289 - 289 \cdot 2 + 81 \cdot 4}{4})}^{\frac{3}{2}}}$

Simplify each term.

Multiply $- 289$ by $2$ .

$\frac{\frac{595}{2}}{{(\frac{289 - 578 + 81 \cdot 4}{4})}^{\frac{3}{2}}}$

Multiply $81$ by $4$ .

$\frac{\frac{595}{2}}{{(\frac{289 - 578 + 324}{4})}^{\frac{3}{2}}}$

Subtract $578$ from $289$ .

$\frac{\frac{595}{2}}{{(\frac{- 289 + 324}{4})}^{\frac{3}{2}}}$

Add $- 289$ and $324$ .

$\frac{\frac{595}{2}}{{(\frac{35}{4})}^{\frac{3}{2}}}$

Apply the product rule to $\frac{35}{4}$ .

$\frac{\frac{595}{2}}{\frac{35^{\frac{3}{2}}}{4^{\frac{3}{2}}}}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$\frac{\frac{595}{2}}{\frac{35^{\frac{3}{2}}}{{(2^{2})}^{\frac{3}{2}}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{595}{2}}{\frac{35^{\frac{3}{2}}}{2^{2 (\frac{3}{2})}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\frac{595}{2}}{\frac{35^{\frac{3}{2}}}{2^{2 (\frac{3}{2})}}}$

Rewrite the expression.

$\frac{\frac{595}{2}}{\frac{35^{\frac{3}{2}}}{2^{3}}}$

Raise $2$ to the power of $3$ .

$\frac{\frac{595}{2}}{\frac{35^{\frac{3}{2}}}{8}}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{595}{2} \cdot \frac{8}{35^{\frac{3}{2}}}$

Combine.

$\frac{595 \cdot 8}{2 \cdot 35^{\frac{3}{2}}}$

Factor $2$ out of $595 \cdot 8$ .

$\frac{2 (595 \cdot 4)}{2 \cdot 35^{\frac{3}{2}}}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 35^{\frac{3}{2}}$ .

$\frac{2 (595 \cdot 4)}{2 (35^{\frac{3}{2}})}$

Cancel the common factor.

$\frac{2 (595 \cdot 4)}{2 \cdot 35^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{595 \cdot 4}{35^{\frac{3}{2}}}$

Multiply $595$ by $4$ .

$\frac{2380}{35^{\frac{3}{2}}}$

Step 15

$x = \frac{\sqrt{34}}{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{\sqrt{34}}{2}$ is a local minimum

Step 16

Find the y-value when $x = \frac{\sqrt{34}}{2}$ .

Replace the variable $x$ with $\frac{\sqrt{34}}{2}$ in the expression.

$f (\frac{\sqrt{34}}{2}) = \sqrt{{(\frac{\sqrt{34}}{2})}^{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81}$

Simplify the result.

Apply the product rule to $\frac{\sqrt{34}}{2}$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{{\sqrt{34}}^{4}}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81}$

Simplify the numerator.

Rewrite ${\sqrt{34}}^{4}$ as $34^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{{(34^{\frac{1}{2}})}^{4}}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{34^{\frac{1}{2} \cdot 4}}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81}$

Combine $\frac{1}{2}$ and $4$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{34^{\frac{4}{2}}}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81}$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{34^{\frac{2 \cdot 2}{2}}}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81}$

Cancel the common factors.

Factor $2$ out of $2$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{34^{\frac{2 \cdot 2}{2 (1)}}}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81}$

Cancel the common factor.

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{34^{\frac{2 \cdot 2}{2 \cdot 1}}}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81}$

Rewrite the expression.

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{34^{\frac{2}{1}}}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81}$

Divide $2$ by $1$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{34^{2}}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81}$

Raise $34$ to the power of $2$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{1156}{2^{4}} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81}$

Simplify terms.

Raise $2$ to the power of $4$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{1156}{16} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81}$

Cancel the common factor of $1156$ and $16$ .

Factor $4$ out of $1156$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{4 (289)}{16} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81}$

Cancel the common factors.

Factor $4$ out of $16$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{4 \cdot 289}{4 \cdot 4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81}$

Cancel the common factor.

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{4 \cdot 289}{4 \cdot 4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81}$

Rewrite the expression.

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 {(\frac{\sqrt{34}}{2})}^{2} + 81}$

Apply the product rule to $\frac{\sqrt{34}}{2}$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{{\sqrt{34}}^{2}}{2^{2}} + 81}$

Rewrite ${\sqrt{34}}^{2}$ as $34$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{{(34^{\frac{1}{2}})}^{2}}{2^{2}} + 81}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{34^{\frac{1}{2} \cdot 2}}{2^{2}} + 81}$

Combine $\frac{1}{2}$ and $2$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{34^{\frac{2}{2}}}{2^{2}} + 81}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{34^{\frac{2}{2}}}{2^{2}} + 81}$

Rewrite the expression.

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{34}{2^{2}} + 81}$

Evaluate the exponent.

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{34}{2^{2}} + 81}$

Raise $2$ to the power of $2$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 (\frac{34}{4}) + 81}$

Cancel the common factor of $34$ and $4$ .

Factor $2$ out of $34$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{2 (17)}{4} + 81}$

Cancel the common factors.

Factor $2$ out of $4$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{2 \cdot 17}{2 \cdot 2} + 81}$

Cancel the common factor.

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{2 \cdot 17}{2 \cdot 2} + 81}$

Rewrite the expression.

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 (\frac{17}{2}) + 81}$

Multiply $- 17 (\frac{17}{2})$ .

Combine $- 17$ and $\frac{17}{2}$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} + \frac{- 17 \cdot 17}{2} + 81}$

Multiply $- 17$ by $17$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} + \frac{- 289}{2} + 81}$

Move the negative in front of the fraction.

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - \frac{289}{2} + 81}$

To write $- \frac{289}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - \frac{289}{2} \cdot \frac{2}{2} + 81}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{289}{2}$ by $\frac{2}{2}$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - \frac{289 \cdot 2}{2 \cdot 2} + 81}$

Multiply $2$ by $2$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - \frac{289 \cdot 2}{4} + 81}$

Combine the numerators over the common denominator.

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289 - 289 \cdot 2}{4} + 81}$

Simplify the numerator.

Multiply $- 289$ by $2$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{289 - 578}{4} + 81}$

Subtract $578$ from $289$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{- 289}{4} + 81}$

To write $81$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{- 289}{4} + 81 \cdot \frac{4}{4}}$

Combine $81$ and $\frac{4}{4}$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{- 289}{4} + \frac{81 \cdot 4}{4}}$

Combine the numerators over the common denominator.

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{- 289 + 81 \cdot 4}{4}}$

Simplify the numerator.

Multiply $81$ by $4$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{- 289 + 324}{4}}$

Add $- 289$ and $324$ .

$f (\frac{\sqrt{34}}{2}) = \sqrt{\frac{35}{4}}$

Rewrite $\sqrt{\frac{35}{4}}$ as $\frac{\sqrt{35}}{\sqrt{4}}$ .

$f (\frac{\sqrt{34}}{2}) = \frac{\sqrt{35}}{\sqrt{4}}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$f (\frac{\sqrt{34}}{2}) = \frac{\sqrt{35}}{\sqrt{2^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$f (\frac{\sqrt{34}}{2}) = \frac{\sqrt{35}}{2}$

The final answer is $\frac{\sqrt{35}}{2}$ .

$y = \frac{\sqrt{35}}{2}$

Step 17

Evaluate the second derivative at $x = - \frac{\sqrt{34}}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{2 {(- \frac{\sqrt{34}}{2})}^{6} - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Step 18

Evaluate the second derivative.

Simplify the numerator.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{\sqrt{34}}{2}$ .

$\frac{2 ({(- 1)}^{6} {(\frac{\sqrt{34}}{2})}^{6}) - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Apply the product rule to $\frac{\sqrt{34}}{2}$ .

$\frac{2 ({(- 1)}^{6} \frac{{\sqrt{34}}^{6}}{2^{6}}) - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Raise $- 1$ to the power of $6$ .

$\frac{2 (1 \frac{{\sqrt{34}}^{6}}{2^{6}}) - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Multiply $\frac{{\sqrt{34}}^{6}}{2^{6}}$ by $1$ .

$\frac{2 \frac{{\sqrt{34}}^{6}}{2^{6}} - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Simplify the numerator.

Rewrite ${\sqrt{34}}^{6}$ as $34^{3}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\frac{2 \frac{{(34^{\frac{1}{2}})}^{6}}{2^{6}} - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 \frac{34^{\frac{1}{2} \cdot 6}}{2^{6}} - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine $\frac{1}{2}$ and $6$ .

$\frac{2 \frac{34^{\frac{6}{2}}}{2^{6}} - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6$ .

$\frac{2 \frac{34^{\frac{2 \cdot 3}{2}}}{2^{6}} - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factors.

Factor $2$ out of $2$ .

$\frac{2 \frac{34^{\frac{2 \cdot 3}{2 (1)}}}{2^{6}} - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

$\frac{2 \frac{34^{\frac{2 \cdot 3}{2 \cdot 1}}}{2^{6}} - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{2 \frac{34^{\frac{3}{1}}}{2^{6}} - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Divide $3$ by $1$ .

$\frac{2 \frac{34^{3}}{2^{6}} - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Raise $34$ to the power of $3$ .

$\frac{2 \frac{39304}{2^{6}} - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Raise $2$ to the power of $6$ .

$\frac{2 (\frac{39304}{64}) - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $2$ .

Factor $2$ out of $64$ .

$\frac{2 \frac{39304}{2 (32)} - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

$\frac{2 \frac{39304}{2 \cdot 32} - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{39304}{32} - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $39304$ and $32$ .

Factor $8$ out of $39304$ .

$\frac{\frac{8 (4913)}{32} - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factors.

Factor $8$ out of $32$ .

$\frac{\frac{8 \cdot 4913}{8 \cdot 4} - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

Rewrite the expression.

$\frac{\frac{4913}{4} - 51 {(- \frac{\sqrt{34}}{2})}^{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{\sqrt{34}}{2}$ .

$\frac{\frac{4913}{4} - 51 ({(- 1)}^{4} {(\frac{\sqrt{34}}{2})}^{4}) + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Apply the product rule to $\frac{\sqrt{34}}{2}$ .

$\frac{\frac{4913}{4} - 51 ({(- 1)}^{4} \frac{{\sqrt{34}}^{4}}{2^{4}}) + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Raise $- 1$ to the power of $4$ .

$\frac{\frac{4913}{4} - 51 (1 \frac{{\sqrt{34}}^{4}}{2^{4}}) + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Multiply $\frac{{\sqrt{34}}^{4}}{2^{4}}$ by $1$ .

$\frac{\frac{4913}{4} - 51 \frac{{\sqrt{34}}^{4}}{2^{4}} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Simplify the numerator.

Rewrite ${\sqrt{34}}^{4}$ as $34^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\frac{\frac{4913}{4} - 51 \frac{{(34^{\frac{1}{2}})}^{4}}{2^{4}} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{4913}{4} - 51 \frac{34^{\frac{1}{2} \cdot 4}}{2^{4}} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine $\frac{1}{2}$ and $4$ .

$\frac{\frac{4913}{4} - 51 \frac{34^{\frac{4}{2}}}{2^{4}} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

$\frac{\frac{4913}{4} - 51 \frac{34^{\frac{2 \cdot 2}{2}}}{2^{4}} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factors.

Factor $2$ out of $2$ .

$\frac{\frac{4913}{4} - 51 \frac{34^{\frac{2 \cdot 2}{2 (1)}}}{2^{4}} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

$\frac{\frac{4913}{4} - 51 \frac{34^{\frac{2 \cdot 2}{2 \cdot 1}}}{2^{4}} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{4913}{4} - 51 \frac{34^{\frac{2}{1}}}{2^{4}} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Divide $2$ by $1$ .

$\frac{\frac{4913}{4} - 51 \frac{34^{2}}{2^{4}} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Raise $34$ to the power of $2$ .

$\frac{\frac{4913}{4} - 51 \frac{1156}{2^{4}} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Raise $2$ to the power of $4$ .

$\frac{\frac{4913}{4} - 51 (\frac{1156}{16}) + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $1156$ and $16$ .

Factor $4$ out of $1156$ .

$\frac{\frac{4913}{4} - 51 \frac{4 (289)}{16} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factors.

Factor $4$ out of $16$ .

$\frac{\frac{4913}{4} - 51 \frac{4 \cdot 289}{4 \cdot 4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

$\frac{\frac{4913}{4} - 51 \frac{4 \cdot 289}{4 \cdot 4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{4913}{4} - 51 (\frac{289}{4}) + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Multiply $- 51 (\frac{289}{4})$ .

Combine $- 51$ and $\frac{289}{4}$ .

$\frac{\frac{4913}{4} + \frac{- 51 \cdot 289}{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Multiply $- 51$ by $289$ .

$\frac{\frac{4913}{4} + \frac{- 14739}{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Move the negative in front of the fraction.

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 {(- \frac{\sqrt{34}}{2})}^{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{\sqrt{34}}{2}$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 ({(- 1)}^{2} {(\frac{\sqrt{34}}{2})}^{2}) - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Apply the product rule to $\frac{\sqrt{34}}{2}$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 ({(- 1)}^{2} \frac{{\sqrt{34}}^{2}}{2^{2}}) - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Raise $- 1$ to the power of $2$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 (1 \frac{{\sqrt{34}}^{2}}{2^{2}}) - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Multiply $\frac{{\sqrt{34}}^{2}}{2^{2}}$ by $1$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 \frac{{\sqrt{34}}^{2}}{2^{2}} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite ${\sqrt{34}}^{2}$ as $34$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 \frac{{(34^{\frac{1}{2}})}^{2}}{2^{2}} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 \frac{34^{\frac{1}{2} \cdot 2}}{2^{2}} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 \frac{34^{\frac{2}{2}}}{2^{2}} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 \frac{34^{\frac{2}{2}}}{2^{2}} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 \frac{34^{1}}{2^{2}} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Evaluate the exponent.

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 \frac{34}{2^{2}} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Raise $2$ to the power of $2$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + 486 (\frac{34}{4}) - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $2$ .

Factor $2$ out of $486$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + 2 (243) \frac{34}{4} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Factor $2$ out of $4$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + 2 \cdot 243 \frac{34}{2 \cdot 2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

$\frac{\frac{4913}{4} - \frac{14739}{4} + 2 \cdot 243 \frac{34}{2 \cdot 2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{4913}{4} - \frac{14739}{4} + 243 (\frac{34}{2}) - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine $243$ and $\frac{34}{2}$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + \frac{243 \cdot 34}{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Multiply $243$ by $34$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + \frac{8262}{2} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Divide $8262$ by $2$ .

$\frac{\frac{4913}{4} - \frac{14739}{4} + 4131 - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine the numerators over the common denominator.

$\frac{\frac{4913 - 14739}{4} + 4131 - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Subtract $14739$ from $4913$ .

$\frac{\frac{- 9826}{4} + 4131 - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

To write $4131$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\frac{- 9826}{4} + 4131 \cdot \frac{4}{4} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine $4131$ and $\frac{4}{4}$ .

$\frac{\frac{- 9826}{4} + \frac{4131 \cdot 4}{4} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine the numerators over the common denominator.

$\frac{\frac{- 9826 + 4131 \cdot 4}{4} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Simplify the numerator.

Multiply $4131$ by $4$ .

$\frac{\frac{- 9826 + 16524}{4} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Add $- 9826$ and $16524$ .

$\frac{\frac{6698}{4} - 1377}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

To write $- 1377$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\frac{6698}{4} - 1377 \cdot \frac{4}{4}}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine $- 1377$ and $\frac{4}{4}$ .

$\frac{\frac{6698}{4} + \frac{- 1377 \cdot 4}{4}}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine the numerators over the common denominator.

$\frac{\frac{6698 - 1377 \cdot 4}{4}}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Simplify the numerator.

Multiply $- 1377$ by $4$ .

$\frac{\frac{6698 - 5508}{4}}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Subtract $5508$ from $6698$ .

$\frac{\frac{1190}{4}}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $1190$ and $4$ .

Factor $2$ out of $1190$ .

$\frac{\frac{2 (595)}{4}}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{\frac{2 \cdot 595}{2 \cdot 2}}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

$\frac{\frac{2 \cdot 595}{2 \cdot 2}}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{595}{2}}{{({(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Simplify the denominator.

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{\sqrt{34}}{2}$ .

$\frac{\frac{595}{2}}{{({(- 1)}^{4} {(\frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Apply the product rule to $\frac{\sqrt{34}}{2}$ .

$\frac{\frac{595}{2}}{{({(- 1)}^{4} \frac{{\sqrt{34}}^{4}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Raise $- 1$ to the power of $4$ .

$\frac{\frac{595}{2}}{{(1 \frac{{\sqrt{34}}^{4}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Multiply $\frac{{\sqrt{34}}^{4}}{2^{4}}$ by $1$ .

$\frac{\frac{595}{2}}{{(\frac{{\sqrt{34}}^{4}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Simplify the numerator.

Rewrite ${\sqrt{34}}^{4}$ as $34^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\frac{\frac{595}{2}}{{(\frac{{(34^{\frac{1}{2}})}^{4}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{595}{2}}{{(\frac{34^{\frac{1}{2} \cdot 4}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Combine $\frac{1}{2}$ and $4$ .

$\frac{\frac{595}{2}}{{(\frac{34^{\frac{4}{2}}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

$\frac{\frac{595}{2}}{{(\frac{34^{\frac{2 \cdot 2}{2}}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factors.

Factor $2$ out of $2$ .

$\frac{\frac{595}{2}}{{(\frac{34^{\frac{2 \cdot 2}{2 (1)}}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

$\frac{\frac{595}{2}}{{(\frac{34^{\frac{2 \cdot 2}{2 \cdot 1}}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{595}{2}}{{(\frac{34^{\frac{2}{1}}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Divide $2$ by $1$ .

$\frac{\frac{595}{2}}{{(\frac{34^{2}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Raise $34$ to the power of $2$ .

$\frac{\frac{595}{2}}{{(\frac{1156}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Raise $2$ to the power of $4$ .

$\frac{\frac{595}{2}}{{(\frac{1156}{16} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $1156$ and $16$ .

Factor $4$ out of $1156$ .

$\frac{\frac{595}{2}}{{(\frac{4 (289)}{16} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factors.

Factor $4$ out of $16$ .

$\frac{\frac{595}{2}}{{(\frac{4 \cdot 289}{4 \cdot 4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

$\frac{\frac{595}{2}}{{(\frac{4 \cdot 289}{4 \cdot 4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81)}^{\frac{3}{2}}}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{\sqrt{34}}{2}$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 ({(- 1)}^{2} {(\frac{\sqrt{34}}{2})}^{2}) + 81)}^{\frac{3}{2}}}$

Apply the product rule to $\frac{\sqrt{34}}{2}$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 ({(- 1)}^{2} \frac{{\sqrt{34}}^{2}}{2^{2}}) + 81)}^{\frac{3}{2}}}$

Raise $- 1$ to the power of $2$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 (1 \frac{{\sqrt{34}}^{2}}{2^{2}}) + 81)}^{\frac{3}{2}}}$

Multiply $\frac{{\sqrt{34}}^{2}}{2^{2}}$ by $1$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{{\sqrt{34}}^{2}}{2^{2}} + 81)}^{\frac{3}{2}}}$

Rewrite ${\sqrt{34}}^{2}$ as $34$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{{(34^{\frac{1}{2}})}^{2}}{2^{2}} + 81)}^{\frac{3}{2}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{34^{\frac{1}{2} \cdot 2}}{2^{2}} + 81)}^{\frac{3}{2}}}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{34^{\frac{2}{2}}}{2^{2}} + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{34^{\frac{2}{2}}}{2^{2}} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{34^{1}}{2^{2}} + 81)}^{\frac{3}{2}}}$

Evaluate the exponent.

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{34}{2^{2}} + 81)}^{\frac{3}{2}}}$

Raise $2$ to the power of $2$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 (\frac{34}{4}) + 81)}^{\frac{3}{2}}}$

Cancel the common factor of $34$ and $4$ .

Factor $2$ out of $34$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{2 (17)}{4} + 81)}^{\frac{3}{2}}}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{2 \cdot 17}{2 \cdot 2} + 81)}^{\frac{3}{2}}}$

Cancel the common factor.

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 \frac{2 \cdot 17}{2 \cdot 2} + 81)}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{\frac{595}{2}}{{(\frac{289}{4} - 17 (\frac{17}{2}) + 81)}^{\frac{3}{2}}}$

Multiply $- 17 (\frac{17}{2})$ .

Combine $- 17$ and $\frac{17}{2}$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} + \frac{- 17 \cdot 17}{2} + 81)}^{\frac{3}{2}}}$

Multiply $- 17$ by $17$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} + \frac{- 289}{2} + 81)}^{\frac{3}{2}}}$

Move the negative in front of the fraction.

$\frac{\frac{595}{2}}{{(\frac{289}{4} - \frac{289}{2} + 81)}^{\frac{3}{2}}}$

Find the common denominator.

Multiply $\frac{289}{2}$ by $\frac{2}{2}$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - (\frac{289}{2} \cdot \frac{2}{2}) + 81)}^{\frac{3}{2}}}$

Multiply $\frac{289}{2}$ by $\frac{2}{2}$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - \frac{289 \cdot 2}{2 \cdot 2} + 81)}^{\frac{3}{2}}}$

Write $81$ as a fraction with denominator $1$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - \frac{289 \cdot 2}{2 \cdot 2} + \frac{81}{1})}^{\frac{3}{2}}}$

Multiply $\frac{81}{1}$ by $\frac{4}{4}$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - \frac{289 \cdot 2}{2 \cdot 2} + \frac{81}{1} \cdot \frac{4}{4})}^{\frac{3}{2}}}$

Multiply $\frac{81}{1}$ by $\frac{4}{4}$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - \frac{289 \cdot 2}{2 \cdot 2} + \frac{81 \cdot 4}{4})}^{\frac{3}{2}}}$

Multiply $2$ by $2$ .

$\frac{\frac{595}{2}}{{(\frac{289}{4} - \frac{289 \cdot 2}{4} + \frac{81 \cdot 4}{4})}^{\frac{3}{2}}}$

Combine the numerators over the common denominator.

$\frac{\frac{595}{2}}{{(\frac{289 - 289 \cdot 2 + 81 \cdot 4}{4})}^{\frac{3}{2}}}$

Simplify each term.

Multiply $- 289$ by $2$ .

$\frac{\frac{595}{2}}{{(\frac{289 - 578 + 81 \cdot 4}{4})}^{\frac{3}{2}}}$

Multiply $81$ by $4$ .

$\frac{\frac{595}{2}}{{(\frac{289 - 578 + 324}{4})}^{\frac{3}{2}}}$

Subtract $578$ from $289$ .

$\frac{\frac{595}{2}}{{(\frac{- 289 + 324}{4})}^{\frac{3}{2}}}$

Add $- 289$ and $324$ .

$\frac{\frac{595}{2}}{{(\frac{35}{4})}^{\frac{3}{2}}}$

Apply the product rule to $\frac{35}{4}$ .

$\frac{\frac{595}{2}}{\frac{35^{\frac{3}{2}}}{4^{\frac{3}{2}}}}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$\frac{\frac{595}{2}}{\frac{35^{\frac{3}{2}}}{{(2^{2})}^{\frac{3}{2}}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{595}{2}}{\frac{35^{\frac{3}{2}}}{2^{2 (\frac{3}{2})}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\frac{595}{2}}{\frac{35^{\frac{3}{2}}}{2^{2 (\frac{3}{2})}}}$

Rewrite the expression.

$\frac{\frac{595}{2}}{\frac{35^{\frac{3}{2}}}{2^{3}}}$

Raise $2$ to the power of $3$ .

$\frac{\frac{595}{2}}{\frac{35^{\frac{3}{2}}}{8}}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{595}{2} \cdot \frac{8}{35^{\frac{3}{2}}}$

Combine.

$\frac{595 \cdot 8}{2 \cdot 35^{\frac{3}{2}}}$

Factor $2$ out of $595 \cdot 8$ .

$\frac{2 (595 \cdot 4)}{2 \cdot 35^{\frac{3}{2}}}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 35^{\frac{3}{2}}$ .

$\frac{2 (595 \cdot 4)}{2 (35^{\frac{3}{2}})}$

Cancel the common factor.

$\frac{2 (595 \cdot 4)}{2 \cdot 35^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{595 \cdot 4}{35^{\frac{3}{2}}}$

Multiply $595$ by $4$ .

$\frac{2380}{35^{\frac{3}{2}}}$

Step 19

$x = - \frac{\sqrt{34}}{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{\sqrt{34}}{2}$ is a local minimum

Step 20

Find the y-value when $x = - \frac{\sqrt{34}}{2}$ .

Replace the variable $x$ with $- \frac{\sqrt{34}}{2}$ in the expression.

$f (- \frac{\sqrt{34}}{2}) = \sqrt{{(- \frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81}$

Simplify the result.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{\sqrt{34}}{2}$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{{(- 1)}^{4} {(\frac{\sqrt{34}}{2})}^{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81}$

Apply the product rule to $\frac{\sqrt{34}}{2}$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{{(- 1)}^{4} (\frac{{\sqrt{34}}^{4}}{2^{4}}) - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81}$

Simplify the expression.

Raise $- 1$ to the power of $4$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{1 (\frac{{\sqrt{34}}^{4}}{2^{4}}) - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81}$

Multiply $\frac{{\sqrt{34}}^{4}}{2^{4}}$ by $1$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{{\sqrt{34}}^{4}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81}$

Simplify the numerator.

Rewrite ${\sqrt{34}}^{4}$ as $34^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{{(34^{\frac{1}{2}})}^{4}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{34^{\frac{1}{2} \cdot 4}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81}$

Combine $\frac{1}{2}$ and $4$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{34^{\frac{4}{2}}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81}$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{34^{\frac{2 \cdot 2}{2}}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81}$

Cancel the common factors.

Factor $2$ out of $2$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{34^{\frac{2 \cdot 2}{2 (1)}}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81}$

Cancel the common factor.

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{34^{\frac{2 \cdot 2}{2 \cdot 1}}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81}$

Rewrite the expression.

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{34^{\frac{2}{1}}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81}$

Divide $2$ by $1$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{34^{2}}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81}$

Raise $34$ to the power of $2$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{1156}{2^{4}} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81}$

Reduce the expression by cancelling the common factors.

Raise $2$ to the power of $4$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{1156}{16} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81}$

Cancel the common factor of $1156$ and $16$ .

Factor $4$ out of $1156$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{4 (289)}{16} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81}$

Cancel the common factors.

Factor $4$ out of $16$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{4 \cdot 289}{4 \cdot 4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81}$

Cancel the common factor.

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{4 \cdot 289}{4 \cdot 4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81}$

Rewrite the expression.

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 {(- \frac{\sqrt{34}}{2})}^{2} + 81}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{\sqrt{34}}{2}$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 ({(- 1)}^{2} {(\frac{\sqrt{34}}{2})}^{2}) + 81}$

Apply the product rule to $\frac{\sqrt{34}}{2}$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 ({(- 1)}^{2} (\frac{{\sqrt{34}}^{2}}{2^{2}})) + 81}$

Simplify the expression.

Raise $- 1$ to the power of $2$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 (1 (\frac{{\sqrt{34}}^{2}}{2^{2}})) + 81}$

Multiply $\frac{{\sqrt{34}}^{2}}{2^{2}}$ by $1$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{{\sqrt{34}}^{2}}{2^{2}} + 81}$

Rewrite ${\sqrt{34}}^{2}$ as $34$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{{(34^{\frac{1}{2}})}^{2}}{2^{2}} + 81}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{34^{\frac{1}{2} \cdot 2}}{2^{2}} + 81}$

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{34^{\frac{2}{2}}}{2^{2}} + 81}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{34^{\frac{2}{2}}}{2^{2}} + 81}$

Rewrite the expression.

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{34}{2^{2}} + 81}$

Evaluate the exponent.

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{34}{2^{2}} + 81}$

Raise $2$ to the power of $2$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 (\frac{34}{4}) + 81}$

Cancel the common factor of $34$ and $4$ .

Factor $2$ out of $34$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{2 (17)}{4} + 81}$

Cancel the common factors.

Factor $2$ out of $4$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{2 \cdot 17}{2 \cdot 2} + 81}$

Cancel the common factor.

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 \frac{2 \cdot 17}{2 \cdot 2} + 81}$

Rewrite the expression.

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - 17 (\frac{17}{2}) + 81}$

Multiply $- 17 (\frac{17}{2})$ .

Combine $- 17$ and $\frac{17}{2}$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} + \frac{- 17 \cdot 17}{2} + 81}$

Multiply $- 17$ by $17$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} + \frac{- 289}{2} + 81}$

Move the negative in front of the fraction.

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - \frac{289}{2} + 81}$

To write $- \frac{289}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - \frac{289}{2} \cdot \frac{2}{2} + 81}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{289}{2}$ by $\frac{2}{2}$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - \frac{289 \cdot 2}{2 \cdot 2} + 81}$

Multiply $2$ by $2$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289}{4} - \frac{289 \cdot 2}{4} + 81}$

Combine the numerators over the common denominator.

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289 - 289 \cdot 2}{4} + 81}$

Simplify the numerator.

Multiply $- 289$ by $2$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{289 - 578}{4} + 81}$

Subtract $578$ from $289$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{- 289}{4} + 81}$

To write $81$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{- 289}{4} + 81 \cdot \frac{4}{4}}$

Combine $81$ and $\frac{4}{4}$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{- 289}{4} + \frac{81 \cdot 4}{4}}$

Combine the numerators over the common denominator.

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{- 289 + 81 \cdot 4}{4}}$

Simplify the numerator.

Multiply $81$ by $4$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{- 289 + 324}{4}}$

Add $- 289$ and $324$ .

$f (- \frac{\sqrt{34}}{2}) = \sqrt{\frac{35}{4}}$

Rewrite $\sqrt{\frac{35}{4}}$ as $\frac{\sqrt{35}}{\sqrt{4}}$ .

$f (- \frac{\sqrt{34}}{2}) = \frac{\sqrt{35}}{\sqrt{4}}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$f (- \frac{\sqrt{34}}{2}) = \frac{\sqrt{35}}{\sqrt{2^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$f (- \frac{\sqrt{34}}{2}) = \frac{\sqrt{35}}{2}$

The final answer is $\frac{\sqrt{35}}{2}$ .

$y = \frac{\sqrt{35}}{2}$

Step 21

These are the local extrema for $f (x) = \sqrt{x^{4} - 17 x^{2} + 81}$ .

$(0, 9)$ is a local maxima

$(\frac{\sqrt{34}}{2}, \frac{\sqrt{35}}{2})$ is a local minima

$(- \frac{\sqrt{34}}{2}, \frac{\sqrt{35}}{2})$ is a local minima

Step 22

Find the Local Maxima and Minima square root of x^4-17x^2+81

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