Precalculus Examples

$(- \frac{1}{5}, (\frac{2 \sqrt{6}}{5}))$

Step 1

To find an exponential function, $f (x) = a^{x}$ , containing the point, set $f (x)$ in the function to the $y$ value $\frac{2 \sqrt{6}}{5}$ of the point, and set $x$ to the $x$ value $- \frac{1}{5}$ of the point.

$\frac{2 \sqrt{6}}{5} = a^{- \frac{1}{5}}$

Step 2

Solve the equation for $a$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $a^{- \frac{1}{5}} = \frac{2 \sqrt{6}}{5}$ .

$a^{- \frac{1}{5}} = \frac{2 \sqrt{6}}{5}$

Step 2.2

Raise each side of the equation to the power of $- 5$ to eliminate the fractional exponent on the left side.

${(a^{- \frac{1}{5}})}^{- 5} = {(\frac{2 \sqrt{6}}{5})}^{- 5}$

Step 2.3

Simplify the exponent.

Tap for more steps...

Step 2.3.1

Simplify the left side.

Tap for more steps...

Step 2.3.1.1

Simplify ${(a^{- \frac{1}{5}})}^{- 5}$ .

Tap for more steps...

Step 2.3.1.1.1

Multiply the exponents in ${(a^{- \frac{1}{5}})}^{- 5}$ .

Tap for more steps...

Step 2.3.1.1.1.1

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

$a^{- \frac{1}{5} \cdot - 5} = {(\frac{2 \sqrt{6}}{5})}^{- 5}$

Step 2.3.1.1.1.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 2.3.1.1.1.2.1

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$a^{\frac{- 1}{5} \cdot - 5} = {(\frac{2 \sqrt{6}}{5})}^{- 5}$

Step 2.3.1.1.1.2.2

Factor $5$ out of $- 5$ .

$a^{\frac{- 1}{5} \cdot (5 (- 1))} = {(\frac{2 \sqrt{6}}{5})}^{- 5}$

Step 2.3.1.1.1.2.3

Cancel the common factor.

$a^{\frac{- 1}{5} \cdot (5 \cdot - 1)} = {(\frac{2 \sqrt{6}}{5})}^{- 5}$

Step 2.3.1.1.1.2.4

Rewrite the expression.

$a^{- 1 \cdot - 1} = {(\frac{2 \sqrt{6}}{5})}^{- 5}$

Step 2.3.1.1.1.3

Multiply $- 1$ by $- 1$ .

$a^{1} = {(\frac{2 \sqrt{6}}{5})}^{- 5}$

Step 2.3.1.1.2

Simplify.

$a = {(\frac{2 \sqrt{6}}{5})}^{- 5}$

Step 2.3.2

Simplify the right side.

Tap for more steps...

Step 2.3.2.1

Simplify ${(\frac{2 \sqrt{6}}{5})}^{- 5}$ .

Tap for more steps...

Step 2.3.2.1.1

Change the sign of the exponent by rewriting the base as its reciprocal.

$a = {(\frac{5}{2 \sqrt{6}})}^{5}$

Step 2.3.2.1.2

Multiply $\frac{5}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$a = {(\frac{5}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})}^{5}$

Step 2.3.2.1.3

Combine and simplify the denominator.

Tap for more steps...

Step 2.3.2.1.3.1

Multiply $\frac{5}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$a = {(\frac{5 \sqrt{6}}{2 \sqrt{6} \sqrt{6}})}^{5}$

Step 2.3.2.1.3.2

Move $\sqrt{6}$ .

$a = {(\frac{5 \sqrt{6}}{2 (\sqrt{6} \sqrt{6})})}^{5}$

Step 2.3.2.1.3.3

Raise $\sqrt{6}$ to the power of $1$ .

$a = {(\frac{5 \sqrt{6}}{2 ({\sqrt{6}}^{1} \sqrt{6})})}^{5}$

Step 2.3.2.1.3.4

Raise $\sqrt{6}$ to the power of $1$ .

$a = {(\frac{5 \sqrt{6}}{2 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})})}^{5}$

Step 2.3.2.1.3.5

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

$a = {(\frac{5 \sqrt{6}}{2 {\sqrt{6}}^{1 + 1}})}^{5}$

Step 2.3.2.1.3.6

Add $1$ and $1$ .

$a = {(\frac{5 \sqrt{6}}{2 {\sqrt{6}}^{2}})}^{5}$

Step 2.3.2.1.3.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Tap for more steps...

Step 2.3.2.1.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$a = {(\frac{5 \sqrt{6}}{2 {(6^{\frac{1}{2}})}^{2}})}^{5}$

Step 2.3.2.1.3.7.2

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

$a = {(\frac{5 \sqrt{6}}{2 \cdot 6^{\frac{1}{2} \cdot 2}})}^{5}$

Step 2.3.2.1.3.7.3

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

$a = {(\frac{5 \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}})}^{5}$

Step 2.3.2.1.3.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.2.1.3.7.4.1

Cancel the common factor.

$a = {(\frac{5 \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}})}^{5}$

Step 2.3.2.1.3.7.4.2

Rewrite the expression.

$a = {(\frac{5 \sqrt{6}}{2 \cdot 6^{1}})}^{5}$

Step 2.3.2.1.3.7.5

Evaluate the exponent.

$a = {(\frac{5 \sqrt{6}}{2 \cdot 6})}^{5}$

Step 2.3.2.1.4

Multiply $2$ by $6$ .

$a = {(\frac{5 \sqrt{6}}{12})}^{5}$

Step 2.3.2.1.5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 2.3.2.1.5.1

Apply the product rule to $\frac{5 \sqrt{6}}{12}$ .

$a = \frac{{(5 \sqrt{6})}^{5}}{12^{5}}$

Step 2.3.2.1.5.2

Apply the product rule to $5 \sqrt{6}$ .

$a = \frac{5^{5} {\sqrt{6}}^{5}}{12^{5}}$

Step 2.3.2.1.6

Simplify the numerator.

Tap for more steps...

Step 2.3.2.1.6.1

Raise $5$ to the power of $5$ .

$a = \frac{3125 {\sqrt{6}}^{5}}{12^{5}}$

Step 2.3.2.1.6.2

Rewrite ${\sqrt{6}}^{5}$ as $\sqrt{6^{5}}$ .

$a = \frac{3125 \sqrt{6^{5}}}{12^{5}}$

Step 2.3.2.1.6.3

Raise $6$ to the power of $5$ .

$a = \frac{3125 \sqrt{7776}}{12^{5}}$

Step 2.3.2.1.6.4

Rewrite $7776$ as $36^{2} \cdot 6$ .

Tap for more steps...

Step 2.3.2.1.6.4.1

Factor $1296$ out of $7776$ .

$a = \frac{3125 \sqrt{1296 (6)}}{12^{5}}$

Step 2.3.2.1.6.4.2

Rewrite $1296$ as $36^{2}$ .

$a = \frac{3125 \sqrt{36^{2} \cdot 6}}{12^{5}}$

Step 2.3.2.1.6.5

Pull terms out from under the radical.

$a = \frac{3125 \cdot 36 \sqrt{6}}{12^{5}}$

Step 2.3.2.1.6.6

Multiply $3125$ by $36$ .

$a = \frac{112500 \sqrt{6}}{12^{5}}$

Step 2.3.2.1.7

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 2.3.2.1.7.1

Raise $12$ to the power of $5$ .

$a = \frac{112500 \sqrt{6}}{248832}$

Step 2.3.2.1.7.2

Cancel the common factor of $112500$ and $248832$ .

Tap for more steps...

Step 2.3.2.1.7.2.1

Factor $36$ out of $112500 \sqrt{6}$ .

$a = \frac{36 (3125 \sqrt{6})}{248832}$

Step 2.3.2.1.7.2.2

Cancel the common factors.

Tap for more steps...

Step 2.3.2.1.7.2.2.1

Factor $36$ out of $248832$ .

$a = \frac{36 (3125 \sqrt{6})}{36 (6912)}$

Step 2.3.2.1.7.2.2.2

Cancel the common factor.

$a = \frac{36 (3125 \sqrt{6})}{36 \cdot 6912}$

Step 2.3.2.1.7.2.2.3

Rewrite the expression.

$a = \frac{3125 \sqrt{6}}{6912}$

Step 3

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

$f (x) = {(\frac{3125 \sqrt{6}}{6912})}^{x}$