Find the Exponential Function (-6,5)

$(- 6, 5)$

Step 1

To find an exponential function, $f (x) = a^{x}$ , containing the point, set $f (x)$ in the function to the $y$ value $5$ of the point, and set $x$ to the $x$ value $- 6$ of the point.

$5 = a^{- 6}$

Step 2

Solve the equation for $a$ .

Rewrite the equation as $a^{- 6} = 5$ .

$a^{- 6} = 5$

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

$\frac{1}{a^{6}} = 5$

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.

$a^{6}, 1$

The LCM of one and any expression is the expression.

$a^{6}$

Multiply each term in $\frac{1}{a^{6}} = 5$ by $a^{6}$ to eliminate the fractions.

Multiply each term in $\frac{1}{a^{6}} = 5$ by $a^{6}$ .

$\frac{1}{a^{6}} a^{6} = 5 a^{6}$

Simplify the left side.

Cancel the common factor of $a^{6}$ .

Cancel the common factor.

$\frac{1}{a^{6}} a^{6} = 5 a^{6}$

Rewrite the expression.

$1 = 5 a^{6}$

Solve the equation.

Rewrite the equation as $5 a^{6} = 1$ .

$5 a^{6} = 1$

Divide each term in $5 a^{6} = 1$ by $5$ and simplify.

Divide each term in $5 a^{6} = 1$ by $5$ .

$\frac{5 a^{6}}{5} = \frac{1}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 a^{6}}{5} = \frac{1}{5}$

Divide $a^{6}$ by $1$ .

$a^{6} = \frac{1}{5}$

Take the 6th root of both sides of the equation to eliminate the exponent on the left side.

$a = \pm \sqrt[6]{\frac{1}{5}}$

Simplify $\pm \sqrt[6]{\frac{1}{5}}$ .

Rewrite $\sqrt[6]{\frac{1}{5}}$ as $\frac{\sqrt[6]{1}}{\sqrt[6]{5}}$ .

$a = \pm \frac{\sqrt[6]{1}}{\sqrt[6]{5}}$

Any root of $1$ is $1$ .

$a = \pm \frac{1}{\sqrt[6]{5}}$

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.

$a = \frac{1}{\sqrt[6]{5}}$

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

$a = - \frac{1}{\sqrt[6]{5}}$

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

$a = \frac{1}{\sqrt[6]{5}}, - \frac{1}{\sqrt[6]{5}}$

Step 3

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

$f (x) = {(\frac{1}{\sqrt[6]{5}})}^{x}, f (x) = {(- \frac{1}{\sqrt[6]{5}})}^{x}$

Find the Exponential Function (-6,5)

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