Algebra Examples

f (x) = {(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}

$f (x) = {(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}$

Step 1

Write

$f (x) = {(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}$ as an equation.

$y = {(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}$

Step 2

Interchange the variables.

$x = {(\frac{y^{7} - 2}{3})}^{\frac{1}{5}}$

Step 3

Solve for

$y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as

${(\frac{y^{7} - 2}{3})}^{\frac{1}{5}} = x$ .

${(\frac{y^{7} - 2}{3})}^{\frac{1}{5}} = x$

Step 3.2

Raise each side of the equation to the power of

$5$ to eliminate the fractional exponent on the left side.

${({(\frac{y^{7} - 2}{3})}^{\frac{1}{5}})}^{5} = x^{5}$

Step 3.3

Simplify the left side.

Tap for more steps...

Step 3.3.1

Simplify

${({(\frac{y^{7} - 2}{3})}^{\frac{1}{5}})}^{5}$ .

Tap for more steps...

Step 3.3.1.1

Multiply the exponents in

${({(\frac{y^{7} - 2}{3})}^{\frac{1}{5}})}^{5}$ .

Tap for more steps...

Step 3.3.1.1.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

${(\frac{y^{7} - 2}{3})}^{\frac{1}{5} \cdot 5} = x^{5}$

Step 3.3.1.1.2

Cancel the common factor of

$5$ .

Tap for more steps...

Step 3.3.1.1.2.1

Cancel the common factor.

${(\frac{y^{7} - 2}{3})}^{\frac{1}{5} \cdot 5} = x^{5}$

Step 3.3.1.1.2.2

Rewrite the expression.

${(\frac{y^{7} - 2}{3})}^{1} = x^{5}$

Step 3.3.1.2

Split the fraction

$\frac{y^{7} - 2}{3}$ into two fractions.

${(\frac{y^{7}}{3} + \frac{- 2}{3})}^{1} = x^{5}$

Step 3.3.1.3

Move the negative in front of the fraction.

${(\frac{y^{7}}{3} - \frac{2}{3})}^{1} = x^{5}$

Step 3.3.1.4

Simplify.

$\frac{y^{7}}{3} - \frac{2}{3} = x^{5}$

Step 3.4

Solve for

$y$ .

Tap for more steps...

Step 3.4.1

Add

$\frac{2}{3}$ to both sides of the equation.

$\frac{y^{7}}{3} = x^{5} + \frac{2}{3}$

Step 3.4.2

Multiply both sides of the equation by

$3$ .

$3 \frac{y^{7}}{3} = 3 (x^{5} + \frac{2}{3})$

Step 3.4.3

Simplify both sides of the equation.

Tap for more steps...

Step 3.4.3.1

Simplify the left side.

Tap for more steps...

Step 3.4.3.1.1

Cancel the common factor of

$3$ .

Tap for more steps...

Step 3.4.3.1.1.1

Cancel the common factor.

$3 \frac{y^{7}}{3} = 3 (x^{5} + \frac{2}{3})$

Step 3.4.3.1.1.2

Rewrite the expression.

$y^{7} = 3 (x^{5} + \frac{2}{3})$

Step 3.4.3.2

Simplify the right side.

Tap for more steps...

Step 3.4.3.2.1

Simplify

$3 (x^{5} + \frac{2}{3})$ .

Tap for more steps...

Step 3.4.3.2.1.1

Apply the distributive property.

$y^{7} = 3 x^{5} + 3 (\frac{2}{3})$

Step 3.4.3.2.1.2

Cancel the common factor of

$3$ .

Tap for more steps...

Step 3.4.3.2.1.2.1

Cancel the common factor.

$y^{7} = 3 x^{5} + 3 (\frac{2}{3})$

Step 3.4.3.2.1.2.2

Rewrite the expression.

$y^{7} = 3 x^{5} + 2$

Step 3.4.4

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

$y = \sqrt[7]{3 x^{5} + 2}$

Step 4

Replace

$y$ with

$f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \sqrt[7]{3 x^{5} + 2}$

Step 5

Verify if

$f^{- 1} (x) = \sqrt[7]{3 x^{5} + 2}$ is the inverse of

$f (x) = {(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}$ .

Tap for more steps...

Step 5.1

To verify the inverse, check if

$f^{- 1} (f (x)) = x$ and

$f (f^{- 1} (x)) = x$ .

Step 5.2

Evaluate

$f^{- 1} (f (x))$ .

Tap for more steps...

Step 5.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 5.2.2

Evaluate

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}})$ by substituting in the value of

$f$ into

$f^{- 1}$ .

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{3 {({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}})}^{5} + 2}$

Step 5.2.3

Multiply the exponents in

${({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}})}^{5}$ .

Tap for more steps...

Step 5.2.3.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{3 {(\frac{x^{7} - 2}{3})}^{\frac{1}{5} \cdot 5} + 2}$

Step 5.2.3.2

Cancel the common factor of

$5$ .

Tap for more steps...

Step 5.2.3.2.1

Cancel the common factor.

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{3 {(\frac{x^{7} - 2}{3})}^{\frac{1}{5} \cdot 5} + 2}$

Step 5.2.3.2.2

Rewrite the expression.

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{3 (\frac{x^{7} - 2}{3}) + 2}$

Step 5.2.4

Cancel the common factor of

$3$ .

Tap for more steps...

Step 5.2.4.1

Cancel the common factor.

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{3 (\frac{x^{7} - 2}{3}) + 2}$

Step 5.2.4.2

Rewrite the expression.

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{x^{7} - 2 + 2}$

Step 5.2.5

Simplify by adding numbers.

Tap for more steps...

Step 5.2.5.1

Add

$- 2$ and

$2$ .

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{x^{7} + 0}$

Step 5.2.5.2

Add

$x^{7}$ and

$0$ .

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{x^{7}}$

Step 5.2.6

Pull terms out from under the radical, assuming real numbers.

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = x$

Step 5.3

Evaluate

$f (f^{- 1} (x))$ .

Tap for more steps...

Step 5.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 5.3.2

Evaluate

$f (\sqrt[7]{3 x^{5} + 2})$ by substituting in the value of

$f^{- 1}$ into

$f$ .

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{{(\sqrt[7]{3 x^{5} + 2})}^{7} - 2}{3})}^{\frac{1}{5}}$

Step 5.3.3

Simplify the numerator.

Tap for more steps...

Step 5.3.3.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[7]{3 x^{5} + 2}$ as

${(3 x^{5} + 2)}^{\frac{1}{7}}$ .

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{{({(3 x^{5} + 2)}^{\frac{1}{7}})}^{7} - 2}{3})}^{\frac{1}{5}}$

Step 5.3.3.2

Multiply the exponents in

${({(3 x^{5} + 2)}^{\frac{1}{7}})}^{7}$ .

Tap for more steps...

Step 5.3.3.2.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{{(3 x^{5} + 2)}^{\frac{1}{7} \cdot 7} - 2}{3})}^{\frac{1}{5}}$

Step 5.3.3.2.2

Cancel the common factor of

$7$ .

Tap for more steps...

Step 5.3.3.2.2.1

Cancel the common factor.

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{{(3 x^{5} + 2)}^{\frac{1}{7} \cdot 7} - 2}{3})}^{\frac{1}{5}}$

Step 5.3.3.2.2.2

Rewrite the expression.

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{(3 x^{5} + 2) - 2}{3})}^{\frac{1}{5}}$

Step 5.3.3.3

Simplify.

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{3 x^{5} + 2 - 2}{3})}^{\frac{1}{5}}$

Step 5.3.3.4

Subtract

$2$ from

$2$ .

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{3 x^{5} + 0}{3})}^{\frac{1}{5}}$

Step 5.3.3.5

Add

$3 x^{5}$ and

$0$ .

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{3 x^{5}}{3})}^{\frac{1}{5}}$

Step 5.3.4

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 5.3.4.1

Cancel the common factor of

$3$ .

Tap for more steps...

Step 5.3.4.1.1

Cancel the common factor.

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{3 x^{5}}{3})}^{\frac{1}{5}}$

Step 5.3.4.1.2

Divide

$x^{5}$ by

$1$ .

$f (\sqrt[7]{3 x^{5} + 2}) = {(x^{5})}^{\frac{1}{5}}$

Step 5.3.4.2

Multiply the exponents in

${(x^{5})}^{\frac{1}{5}}$ .

Tap for more steps...

Step 5.3.4.2.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$f (\sqrt[7]{3 x^{5} + 2}) = x^{5 (\frac{1}{5})}$

Step 5.3.4.2.2

Cancel the common factor of

$5$ .

Tap for more steps...

Step 5.3.4.2.2.1

Cancel the common factor.

$f (\sqrt[7]{3 x^{5} + 2}) = x^{5 (\frac{1}{5})}$

Step 5.3.4.2.2.2

Rewrite the expression.

$f (\sqrt[7]{3 x^{5} + 2}) = x$

Step 5.4

Since

$f^{- 1} (f (x)) = x$ and

$f (f^{- 1} (x)) = x$ , then

$f^{- 1} (x) = \sqrt[7]{3 x^{5} + 2}$ is the inverse of

$f (x) = {(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}$ .

$f^{- 1} (x) = \sqrt[7]{3 x^{5} + 2}$