Pre-Algebra Examples

$(- 7, - 3)$

Step 1

To find an exponential function, $f (x) = a^{x}$ , containing the point, set $f (x)$ in the function to the $y$ value $- 3$ of the point, and set $x$ to the $x$ value $- 7$ of the point.

$- 3 = a^{- 7}$

Step 2

Solve the equation for $a$ .

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Step 2.1

Rewrite the equation as $a^{- 7} = - 3$ .

$a^{- 7} = - 3$

Step 2.2

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

$\frac{1}{a^{7}} = - 3$

Step 2.3

Find the LCD of the terms in the equation.

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Step 2.3.1

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

$a^{7}, 1$

Step 2.3.2

The LCM of one and any expression is the expression.

$a^{7}$

Step 2.4

Multiply each term in $\frac{1}{a^{7}} = - 3$ by $a^{7}$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $\frac{1}{a^{7}} = - 3$ by $a^{7}$ .

$\frac{1}{a^{7}} a^{7} = - 3 a^{7}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $a^{7}$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{1}{a^{7}} a^{7} = - 3 a^{7}$

Step 2.4.2.1.2

Rewrite the expression.

$1 = - 3 a^{7}$

Step 2.5

Solve the equation.

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Step 2.5.1

Rewrite the equation as $- 3 a^{7} = 1$ .

$- 3 a^{7} = 1$

Step 2.5.2

Divide each term in $- 3 a^{7} = 1$ by $- 3$ and simplify.

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Step 2.5.2.1

Divide each term in $- 3 a^{7} = 1$ by $- 3$ .

$\frac{- 3 a^{7}}{- 3} = \frac{1}{- 3}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $- 3$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{- 3 a^{7}}{- 3} = \frac{1}{- 3}$

Step 2.5.2.2.1.2

Divide $a^{7}$ by $1$ .

$a^{7} = \frac{1}{- 3}$

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Move the negative in front of the fraction.

$a^{7} = - \frac{1}{3}$

Step 2.5.3

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

$a = \sqrt[7]{- \frac{1}{3}}$

Step 2.5.4

Simplify $\sqrt[7]{- \frac{1}{3}}$ .

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Step 2.5.4.1

Rewrite $- \frac{1}{3}$ as ${({(- 1)}^{7})}^{7} \frac{1}{3}$ .

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Step 2.5.4.1.1

Rewrite $- 1$ as ${(- 1)}^{7}$ .

$a = \sqrt[7]{{(- 1)}^{7} \frac{1}{3}}$

Step 2.5.4.1.2

Rewrite $- 1$ as ${(- 1)}^{7}$ .

$a = \sqrt[7]{{({(- 1)}^{7})}^{7} \frac{1}{3}}$

Step 2.5.4.2

Pull terms out from under the radical.

$a = {(- 1)}^{7} \sqrt[7]{\frac{1}{3}}$

Step 2.5.4.3

Raise $- 1$ to the power of $7$ .

$a = - \sqrt[7]{\frac{1}{3}}$

Step 2.5.4.4

Rewrite $\sqrt[7]{\frac{1}{3}}$ as $\frac{\sqrt[7]{1}}{\sqrt[7]{3}}$ .

$a = - \frac{\sqrt[7]{1}}{\sqrt[7]{3}}$

Step 2.5.4.5

Any root of $1$ is $1$ .

$a = - \frac{1}{\sqrt[7]{3}}$

Step 3

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

$f (x) = {(- \frac{1}{\sqrt[7]{3}})}^{x}$