Write as a Function of a a=( square root of 3)/4(s)^2

$a = \frac{\sqrt{3}}{4} {(s)}^{2}$

Step 1

Rewrite the equation as $\frac{\sqrt{3}}{4} s^{2} = a$ .

$\frac{\sqrt{3}}{4} s^{2} = a$

Step 2

Multiply both sides of the equation by $\frac{4}{\sqrt{3}}$ .

$\frac{4}{\sqrt{3}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Step 3

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{4}{\sqrt{3}} (\frac{\sqrt{3}}{4} s^{2})$ .

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{4}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Combine and simplify the denominator.

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{4 \sqrt{3}}{\sqrt{3} \sqrt{3}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{4 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{4 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

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

$\frac{4 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Add $1$ and $1$ .

$\frac{4 \sqrt{3}}{{\sqrt{3}}^{2}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{4 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

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

$\frac{4 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

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

$\frac{4 \sqrt{3}}{3^{\frac{2}{2}}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4 \sqrt{3}}{3^{\frac{2}{2}}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Rewrite the expression.

$\frac{4 \sqrt{3}}{3^{1}} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Evaluate the exponent.

$\frac{4 \sqrt{3}}{3} (\frac{\sqrt{3}}{4} s^{2}) = \frac{4}{\sqrt{3}} a$

Combine $\frac{\sqrt{3}}{4}$ and $s^{2}$ .

$\frac{4 \sqrt{3}}{3} \cdot \frac{\sqrt{3} s^{2}}{4} = \frac{4}{\sqrt{3}} a$

Combine.

$\frac{4 \sqrt{3} (\sqrt{3} s^{2})}{3 \cdot 4} = \frac{4}{\sqrt{3}} a$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 \sqrt{3} (\sqrt{3} s^{2})}{3 \cdot 4} = \frac{4}{\sqrt{3}} a$

Rewrite the expression.

$\frac{\sqrt{3} (\sqrt{3} s^{2})}{3} = \frac{4}{\sqrt{3}} a$

Multiply $\sqrt{3}$ by $\sqrt{3}$ .

$\frac{{\sqrt{3}}^{2} s^{2}}{3} = \frac{4}{\sqrt{3}} a$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{{(3^{\frac{1}{2}})}^{2} s^{2}}{3} = \frac{4}{\sqrt{3}} a$

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

$\frac{3^{\frac{1}{2} \cdot 2} s^{2}}{3} = \frac{4}{\sqrt{3}} a$

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

$\frac{3^{\frac{2}{2}} s^{2}}{3} = \frac{4}{\sqrt{3}} a$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3^{\frac{2}{2}} s^{2}}{3} = \frac{4}{\sqrt{3}} a$

Rewrite the expression.

$\frac{3^{1} s^{2}}{3} = \frac{4}{\sqrt{3}} a$

Evaluate the exponent.

$\frac{3 s^{2}}{3} = \frac{4}{\sqrt{3}} a$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 s^{2}}{3} = \frac{4}{\sqrt{3}} a$

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

$s^{2} = \frac{4}{\sqrt{3}} a$

Simplify the right side.

Simplify $\frac{4}{\sqrt{3}} a$ .

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$s^{2} = \frac{4}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} a$

Combine and simplify the denominator.

Multiply $\frac{4}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$s^{2} = \frac{4 \sqrt{3}}{\sqrt{3} \sqrt{3}} a$

Raise $\sqrt{3}$ to the power of $1$ .

$s^{2} = \frac{4 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} a$

Raise $\sqrt{3}$ to the power of $1$ .

$s^{2} = \frac{4 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} a$

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

$s^{2} = \frac{4 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} a$

Add $1$ and $1$ .

$s^{2} = \frac{4 \sqrt{3}}{{\sqrt{3}}^{2}} a$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$s^{2} = \frac{4 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} a$

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

$s^{2} = \frac{4 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} a$

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

$s^{2} = \frac{4 \sqrt{3}}{3^{\frac{2}{2}}} a$

Cancel the common factor of $2$ .

Cancel the common factor.

$s^{2} = \frac{4 \sqrt{3}}{3^{\frac{2}{2}}} a$

Rewrite the expression.

$s^{2} = \frac{4 \sqrt{3}}{3^{1}} a$

Evaluate the exponent.

$s^{2} = \frac{4 \sqrt{3}}{3} a$

Combine $\frac{4 \sqrt{3}}{3}$ and $a$ .

$s^{2} = \frac{4 \sqrt{3} a}{3}$

Step 4

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

$s = \pm \sqrt{\frac{4 \sqrt{3} a}{3}}$

Step 5

Simplify $\pm \sqrt{\frac{4 \sqrt{3} a}{3}}$ .

Rewrite $\sqrt{\frac{4 \sqrt{3} a}{3}}$ as $\frac{\sqrt{4 \sqrt{3} a}}{\sqrt{3}}$ .

$s = \pm \frac{\sqrt{4 \sqrt{3} a}}{\sqrt{3}}$

Simplify the numerator.

Rewrite $4 \sqrt{3} a$ as $2^{2} (\sqrt{3} a)$ .

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

$s = \pm \frac{\sqrt{2^{2} \sqrt{3} a}}{\sqrt{3}}$

Add parentheses.

$s = \pm \frac{\sqrt{2^{2} (\sqrt{3} a)}}{\sqrt{3}}$

Pull terms out from under the radical.

$s = \pm \frac{2 \sqrt{\sqrt{3} a}}{\sqrt{3}}$

Multiply $\frac{2 \sqrt{\sqrt{3} a}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$s = \pm \frac{2 \sqrt{\sqrt{3} a}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Combine and simplify the denominator.

Multiply $\frac{2 \sqrt{\sqrt{3} a}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

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

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{{\sqrt{3}}^{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

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

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{3^{\frac{2}{2}}}$

Rewrite the expression.

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{3^{1}}$

Evaluate the exponent.

$s = \pm \frac{2 \sqrt{\sqrt{3} a} \sqrt{3}}{3}$

Combine using the product rule for radicals.

$s = \pm \frac{2 \sqrt{3 \sqrt{3} a}}{3}$

Step 6

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.

$s = \frac{2 \sqrt{3 \sqrt{3} a}}{3}$

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

$s = - \frac{2 \sqrt{3 \sqrt{3} a}}{3}$

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

$s = \frac{2 \sqrt{3 \sqrt{3} a}}{3}$

$s = - \frac{2 \sqrt{3 \sqrt{3} a}}{3}$

$s = \frac{2 \sqrt{3 \sqrt{3} a}}{3}$

$s = - \frac{2 \sqrt{3 \sqrt{3} a}}{3}$

Step 7

To rewrite as a function of $a$ , write the equation so that $s$ is by itself on one side of the equal sign and an expression involving only $a$ is on the other side.

$f (a) = \frac{2 \sqrt{3 \sqrt{3} a}}{3}, - \frac{2 \sqrt{3 \sqrt{3} a}}{3}$

Write as a Function of a a=( square root of 3)/4(s)^2

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