Finite Math Examples

$(- 4, 8)$

Step 1

To find an exponential function, $f (x) = a^{x}$ , containing the point, set $f (x)$ in the function to the $y$ value $8$ of the point, and set $x$ to the $x$ value $- 4$ of the point.

$8 = a^{- 4}$

Step 2

Solve the equation for $a$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $a^{- 4} = 8$ .

$a^{- 4} = 8$

Step 2.2

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

$\frac{1}{a^{4}} = 8$

Step 2.3

Find the LCD of the terms in the equation.

Tap for more steps...

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.

$a^{4}, 1$

Step 2.3.2

The LCM of one and any expression is the expression.

$a^{4}$

Step 2.4

Multiply each term in $\frac{1}{a^{4}} = 8$ by $a^{4}$ to eliminate the fractions.

Tap for more steps...

Step 2.4.1

Multiply each term in $\frac{1}{a^{4}} = 8$ by $a^{4}$ .

$\frac{1}{a^{4}} a^{4} = 8 a^{4}$

Step 2.4.2

Simplify the left side.

Tap for more steps...

Step 2.4.2.1

Cancel the common factor of $a^{4}$ .

Tap for more steps...

Step 2.4.2.1.1

Cancel the common factor.

$\frac{1}{a^{4}} a^{4} = 8 a^{4}$

Step 2.4.2.1.2

Rewrite the expression.

$1 = 8 a^{4}$

Step 2.5

Solve the equation.

Tap for more steps...

Step 2.5.1

Rewrite the equation as $8 a^{4} = 1$ .

$8 a^{4} = 1$

Step 2.5.2

Divide each term in $8 a^{4} = 1$ by $8$ and simplify.

Tap for more steps...

Step 2.5.2.1

Divide each term in $8 a^{4} = 1$ by $8$ .

$\frac{8 a^{4}}{8} = \frac{1}{8}$

Step 2.5.2.2

Simplify the left side.

Tap for more steps...

Step 2.5.2.2.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{8 a^{4}}{8} = \frac{1}{8}$

Step 2.5.2.2.1.2

Divide $a^{4}$ by $1$ .

$a^{4} = \frac{1}{8}$

Step 2.5.3

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

$a = \pm \sqrt[4]{\frac{1}{8}}$

Step 2.5.4

Simplify $\pm \sqrt[4]{\frac{1}{8}}$ .

Tap for more steps...

Step 2.5.4.1

Rewrite $\sqrt[4]{\frac{1}{8}}$ as $\frac{\sqrt[4]{1}}{\sqrt[4]{8}}$ .

$a = \pm \frac{\sqrt[4]{1}}{\sqrt[4]{8}}$

Step 2.5.4.2

Any root of $1$ is $1$ .

$a = \pm \frac{1}{\sqrt[4]{8}}$

Step 2.5.4.3

Multiply $\frac{1}{\sqrt[4]{8}}$ by $\frac{{\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{3}}$ .

$a = \pm \frac{1}{\sqrt[4]{8}} \cdot \frac{{\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{3}}$

Step 2.5.4.4

Combine and simplify the denominator.

Tap for more steps...

Step 2.5.4.4.1

Multiply $\frac{1}{\sqrt[4]{8}}$ by $\frac{{\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{3}}$ .

$a = \pm \frac{{\sqrt[4]{8}}^{3}}{\sqrt[4]{8} {\sqrt[4]{8}}^{3}}$

Step 2.5.4.4.2

Raise $\sqrt[4]{8}$ to the power of $1$ .

$a = \pm \frac{{\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{1} {\sqrt[4]{8}}^{3}}$

Step 2.5.4.4.3

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

$a = \pm \frac{{\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{1 + 3}}$

Step 2.5.4.4.4

Add $1$ and $3$ .

$a = \pm \frac{{\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{4}}$

Step 2.5.4.4.5

Rewrite ${\sqrt[4]{8}}^{4}$ as $8$ .

Tap for more steps...

Step 2.5.4.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{8}$ as $8^{\frac{1}{4}}$ .

$a = \pm \frac{{\sqrt[4]{8}}^{3}}{{(8^{\frac{1}{4}})}^{4}}$

Step 2.5.4.4.5.2

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

$a = \pm \frac{{\sqrt[4]{8}}^{3}}{8^{\frac{1}{4} \cdot 4}}$

Step 2.5.4.4.5.3

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

$a = \pm \frac{{\sqrt[4]{8}}^{3}}{8^{\frac{4}{4}}}$

Step 2.5.4.4.5.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.5.4.4.5.4.1

Cancel the common factor.

$a = \pm \frac{{\sqrt[4]{8}}^{3}}{8^{\frac{4}{4}}}$

Step 2.5.4.4.5.4.2

Rewrite the expression.

$a = \pm \frac{{\sqrt[4]{8}}^{3}}{8^{1}}$

Step 2.5.4.4.5.5

Evaluate the exponent.

$a = \pm \frac{{\sqrt[4]{8}}^{3}}{8}$

Step 2.5.4.5

Simplify the numerator.

Tap for more steps...

Step 2.5.4.5.1

Rewrite ${\sqrt[4]{8}}^{3}$ as $\sqrt[4]{8^{3}}$ .

$a = \pm \frac{\sqrt[4]{8^{3}}}{8}$

Step 2.5.4.5.2

Raise $8$ to the power of $3$ .

$a = \pm \frac{\sqrt[4]{512}}{8}$

Step 2.5.4.5.3

Rewrite $512$ as $4^{4} \cdot 2$ .

Tap for more steps...

Step 2.5.4.5.3.1

Factor $256$ out of $512$ .

$a = \pm \frac{\sqrt[4]{256 (2)}}{8}$

Step 2.5.4.5.3.2

Rewrite $256$ as $4^{4}$ .

$a = \pm \frac{\sqrt[4]{4^{4} \cdot 2}}{8}$

Step 2.5.4.5.4

Pull terms out from under the radical.

$a = \pm \frac{4 \sqrt[4]{2}}{8}$

Step 2.5.4.6

Cancel the common factor of $4$ and $8$ .

Tap for more steps...

Step 2.5.4.6.1

Factor $4$ out of $4 \sqrt[4]{2}$ .

$a = \pm \frac{4 (\sqrt[4]{2})}{8}$

Step 2.5.4.6.2

Cancel the common factors.

Tap for more steps...

Step 2.5.4.6.2.1

Factor $4$ out of $8$ .

$a = \pm \frac{4 \sqrt[4]{2}}{4 \cdot 2}$

Step 2.5.4.6.2.2

Cancel the common factor.

$a = \pm \frac{4 \sqrt[4]{2}}{4 \cdot 2}$

Step 2.5.4.6.2.3

Rewrite the expression.

$a = \pm \frac{\sqrt[4]{2}}{2}$

Step 2.5.5

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

Tap for more steps...

Step 2.5.5.1

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

$a = \frac{\sqrt[4]{2}}{2}$

Step 2.5.5.2

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

$a = - \frac{\sqrt[4]{2}}{2}$

Step 2.5.5.3

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

$a = \frac{\sqrt[4]{2}}{2}, - \frac{\sqrt[4]{2}}{2}$

Step 3

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

$f (x) = {(\frac{\sqrt[4]{2}}{2})}^{x}, f (x) = {(- \frac{\sqrt[4]{2}}{2})}^{x}$