Find the Exponential Function (-5,12)

$(- 5, 12)$

Step 1

To find an exponential function, $f (x) = a^{x}$ , containing the point, set $f (x)$ in the function to the $y$ value $12$ of the point, and set $x$ to the $x$ value $- 5$ of the point.

$12 = a^{- 5}$

Step 2

Solve the equation for $a$ .

Rewrite the equation as $a^{- 5} = 12$ .

$a^{- 5} = 12$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{a^{5}} = 12$

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$a^{5}, 1$

The LCM of one and any expression is the expression.

$a^{5}$

Multiply each term in $\frac{1}{a^{5}} = 12$ by $a^{5}$ to eliminate the fractions.

Multiply each term in $\frac{1}{a^{5}} = 12$ by $a^{5}$ .

$\frac{1}{a^{5}} a^{5} = 12 a^{5}$

Simplify the left side.

Cancel the common factor of $a^{5}$ .

Cancel the common factor.

$\frac{1}{a^{5}} a^{5} = 12 a^{5}$

Rewrite the expression.

$1 = 12 a^{5}$

Solve the equation.

Rewrite the equation as $12 a^{5} = 1$ .

$12 a^{5} = 1$

Divide each term in $12 a^{5} = 1$ by $12$ and simplify.

Divide each term in $12 a^{5} = 1$ by $12$ .

$\frac{12 a^{5}}{12} = \frac{1}{12}$

Simplify the left side.

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12 a^{5}}{12} = \frac{1}{12}$

Divide $a^{5}$ by $1$ .

$a^{5} = \frac{1}{12}$

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

$a = \sqrt[5]{\frac{1}{12}}$

Simplify $\sqrt[5]{\frac{1}{12}}$ .

Rewrite $\sqrt[5]{\frac{1}{12}}$ as $\frac{\sqrt[5]{1}}{\sqrt[5]{12}}$ .

$a = \frac{\sqrt[5]{1}}{\sqrt[5]{12}}$

Any root of $1$ is $1$ .

$a = \frac{1}{\sqrt[5]{12}}$

Multiply $\frac{1}{\sqrt[5]{12}}$ by $\frac{{\sqrt[5]{12}}^{4}}{{\sqrt[5]{12}}^{4}}$ .

$a = \frac{1}{\sqrt[5]{12}} \cdot \frac{{\sqrt[5]{12}}^{4}}{{\sqrt[5]{12}}^{4}}$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt[5]{12}}$ by $\frac{{\sqrt[5]{12}}^{4}}{{\sqrt[5]{12}}^{4}}$ .

$a = \frac{{\sqrt[5]{12}}^{4}}{\sqrt[5]{12} {\sqrt[5]{12}}^{4}}$

Raise $\sqrt[5]{12}$ to the power of $1$ .

$a = \frac{{\sqrt[5]{12}}^{4}}{{\sqrt[5]{12}}^{1} {\sqrt[5]{12}}^{4}}$

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

$a = \frac{{\sqrt[5]{12}}^{4}}{{\sqrt[5]{12}}^{1 + 4}}$

Add $1$ and $4$ .

$a = \frac{{\sqrt[5]{12}}^{4}}{{\sqrt[5]{12}}^{5}}$

Rewrite ${\sqrt[5]{12}}^{5}$ as $12$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{12}$ as $12^{\frac{1}{5}}$ .

$a = \frac{{\sqrt[5]{12}}^{4}}{{(12^{\frac{1}{5}})}^{5}}$

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

$a = \frac{{\sqrt[5]{12}}^{4}}{12^{\frac{1}{5} \cdot 5}}$

Combine $\frac{1}{5}$ and $5$ .

$a = \frac{{\sqrt[5]{12}}^{4}}{12^{\frac{5}{5}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$a = \frac{{\sqrt[5]{12}}^{4}}{12^{\frac{5}{5}}}$

Rewrite the expression.

$a = \frac{{\sqrt[5]{12}}^{4}}{12^{1}}$

Evaluate the exponent.

$a = \frac{{\sqrt[5]{12}}^{4}}{12}$

Simplify the numerator.

Rewrite ${\sqrt[5]{12}}^{4}$ as $\sqrt[5]{12^{4}}$ .

$a = \frac{\sqrt[5]{12^{4}}}{12}$

Raise $12$ to the power of $4$ .

$a = \frac{\sqrt[5]{20736}}{12}$

Rewrite $20736$ as $2^{5} \cdot 648$ .

Factor $32$ out of $20736$ .

$a = \frac{\sqrt[5]{32 (648)}}{12}$

Rewrite $32$ as $2^{5}$ .

$a = \frac{\sqrt[5]{2^{5} \cdot 648}}{12}$

Pull terms out from under the radical.

$a = \frac{2 \sqrt[5]{648}}{12}$

Cancel the common factor of $2$ and $12$ .

Factor $2$ out of $2 \sqrt[5]{648}$ .

$a = \frac{2 (\sqrt[5]{648})}{12}$

Cancel the common factors.

Factor $2$ out of $12$ .

$a = \frac{2 \sqrt[5]{648}}{2 \cdot 6}$

Cancel the common factor.

$a = \frac{2 \sqrt[5]{648}}{2 \cdot 6}$

Rewrite the expression.

$a = \frac{\sqrt[5]{648}}{6}$

Step 3

Substitute each value for $a$ back into the function $f (x) = a^{x}$ to find each possible exponential function.

$f (x) = {(\frac{\sqrt[5]{648}}{6})}^{x}$

Find the Exponential Function (-5,12)

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