Algebra Examples

$f (x) = \frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9}$

Step 1

Write $f (x) = \frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9}$ as an equation.

$y = \frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9}$

Step 2

Interchange the variables.

$x = \frac{{(y^{7} + 10)}^{\frac{1}{5}}}{9}$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $\frac{{(y^{7} + 10)}^{\frac{1}{5}}}{9} = x$ .

$\frac{{(y^{7} + 10)}^{\frac{1}{5}}}{9} = x$

Step 3.2

Multiply both sides of the equation by $9$ .

$9 \frac{{(y^{7} + 10)}^{\frac{1}{5}}}{9} = 9 x$

Step 3.3

Simplify the left side.

Tap for more steps...

Step 3.3.1

Cancel the common factor of $9$ .

Tap for more steps...

Step 3.3.1.1

Cancel the common factor.

$9 \frac{{(y^{7} + 10)}^{\frac{1}{5}}}{9} = 9 x$

Step 3.3.1.2

Rewrite the expression.

${(y^{7} + 10)}^{\frac{1}{5}} = 9 x$

Step 3.4

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

${({(y^{7} + 10)}^{\frac{1}{5}})}^{5} = {(9 x)}^{5}$

Step 3.5

Simplify the exponent.

Tap for more steps...

Step 3.5.1

Simplify the left side.

Tap for more steps...

Step 3.5.1.1

Simplify ${({(y^{7} + 10)}^{\frac{1}{5}})}^{5}$ .

Tap for more steps...

Step 3.5.1.1.1

Multiply the exponents in ${({(y^{7} + 10)}^{\frac{1}{5}})}^{5}$ .

Tap for more steps...

Step 3.5.1.1.1.1

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

${(y^{7} + 10)}^{\frac{1}{5} \cdot 5} = {(9 x)}^{5}$

Step 3.5.1.1.1.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.5.1.1.1.2.1

Cancel the common factor.

${(y^{7} + 10)}^{\frac{1}{5} \cdot 5} = {(9 x)}^{5}$

Step 3.5.1.1.1.2.2

Rewrite the expression.

${(y^{7} + 10)}^{1} = {(9 x)}^{5}$

Step 3.5.1.1.2

Simplify.

$y^{7} + 10 = {(9 x)}^{5}$

Step 3.5.2

Simplify the right side.

Tap for more steps...

Step 3.5.2.1

Simplify ${(9 x)}^{5}$ .

Tap for more steps...

Step 3.5.2.1.1

Apply the product rule to $9 x$ .

$y^{7} + 10 = 9^{5} x^{5}$

Step 3.5.2.1.2

Raise $9$ to the power of $5$ .

$y^{7} + 10 = 59049 x^{5}$

Step 3.6

Solve for $y$ .

Tap for more steps...

Step 3.6.1

Subtract $10$ from both sides of the equation.

$y^{7} = 59049 x^{5} - 10$

Step 3.6.2

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

$y = \sqrt[7]{59049 x^{5} - 10}$

Step 4

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

$f^{- 1} (x) = \sqrt[7]{59049 x^{5} - 10}$

Step 5

Verify if $f^{- 1} (x) = \sqrt[7]{59049 x^{5} - 10}$ is the inverse of $f (x) = \frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9}$ .

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 $f^{- 1} (\frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9}) = \sqrt[7]{59049 {(\frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9})}^{5} - 10}$

Step 5.2.3

Apply the product rule to $\frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9}$ .

$f^{- 1} (\frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9}) = \sqrt[7]{59049 (\frac{{({(x^{7} + 10)}^{\frac{1}{5}})}^{5}}{9^{5}}) - 10}$

Step 5.2.4

Simplify the numerator.

Tap for more steps...

Step 5.2.4.1

Multiply the exponents in ${({(x^{7} + 10)}^{\frac{1}{5}})}^{5}$ .

Tap for more steps...

Step 5.2.4.1.1

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

$f^{- 1} (\frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9}) = \sqrt[7]{59049 (\frac{{(x^{7} + 10)}^{\frac{1}{5} \cdot 5}}{9^{5}}) - 10}$

Step 5.2.4.1.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.2.4.1.2.1

Cancel the common factor.

$f^{- 1} (\frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9}) = \sqrt[7]{59049 (\frac{{(x^{7} + 10)}^{\frac{1}{5} \cdot 5}}{9^{5}}) - 10}$

Step 5.2.4.1.2.2

Rewrite the expression.

$f^{- 1} (\frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9}) = \sqrt[7]{59049 (\frac{x^{7} + 10}{9^{5}}) - 10}$

Step 5.2.4.2

Simplify.

$f^{- 1} (\frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9}) = \sqrt[7]{59049 (\frac{x^{7} + 10}{9^{5}}) - 10}$

Step 5.2.5

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 5.2.5.1

Raise $9$ to the power of $5$ .

$f^{- 1} (\frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9}) = \sqrt[7]{59049 (\frac{x^{7} + 10}{59049}) - 10}$

Step 5.2.5.2

Cancel the common factor of $59049$ .

Tap for more steps...

Step 5.2.5.2.1

Cancel the common factor.

$f^{- 1} (\frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9}) = \sqrt[7]{59049 (\frac{x^{7} + 10}{59049}) - 10}$

Step 5.2.5.2.2

Rewrite the expression.

$f^{- 1} (\frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9}) = \sqrt[7]{x^{7} + 10 - 10}$

Step 5.2.5.3

Simplify by subtracting numbers.

Tap for more steps...

Step 5.2.5.3.1

Subtract $10$ from $10$ .

$f^{- 1} (\frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9}) = \sqrt[7]{x^{7} + 0}$

Step 5.2.5.3.2

Add $x^{7}$ and $0$ .

$f^{- 1} (\frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9}) = \sqrt[7]{x^{7}}$

Step 5.2.6

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

$f^{- 1} (\frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9}) = x$

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 $f (\sqrt[7]{59049 x^{5} - 10})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{{({(\sqrt[7]{59049 x^{5} - 10})}^{7} + 10)}^{\frac{1}{5}}}{9}$

Step 5.3.3

Simplify the numerator.

Tap for more steps...

Step 5.3.3.1

Rewrite ${\sqrt[7]{59049 x^{5} - 10}}^{7}$ as $59049 x^{5} - 10$ .

Tap for more steps...

Step 5.3.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{59049 x^{5} - 10}$ as ${(59049 x^{5} - 10)}^{\frac{1}{7}}$ .

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{{({({(59049 x^{5} - 10)}^{\frac{1}{7}})}^{7} + 10)}^{\frac{1}{5}}}{9}$

Step 5.3.3.1.2

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

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{{({(59049 x^{5} - 10)}^{\frac{1}{7} \cdot 7} + 10)}^{\frac{1}{5}}}{9}$

Step 5.3.3.1.3

Combine $\frac{1}{7}$ and $7$ .

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{{({(59049 x^{5} - 10)}^{\frac{7}{7}} + 10)}^{\frac{1}{5}}}{9}$

Step 5.3.3.1.4

Cancel the common factor of $7$ .

Tap for more steps...

Step 5.3.3.1.4.1

Cancel the common factor.

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{{({(59049 x^{5} - 10)}^{\frac{7}{7}} + 10)}^{\frac{1}{5}}}{9}$

Step 5.3.3.1.4.2

Rewrite the expression.

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{{((59049 x^{5} - 10) + 10)}^{\frac{1}{5}}}{9}$

Step 5.3.3.1.5

Simplify.

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{{(59049 x^{5} - 10 + 10)}^{\frac{1}{5}}}{9}$

Step 5.3.3.2

Combine the opposite terms in $59049 x^{5} - 10 + 10$ .

Tap for more steps...

Step 5.3.3.2.1

Add $- 10$ and $10$ .

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{{(59049 x^{5} + 0)}^{\frac{1}{5}}}{9}$

Step 5.3.3.2.2

Add $59049 x^{5}$ and $0$ .

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{{(59049 x^{5})}^{\frac{1}{5}}}{9}$

Step 5.3.3.3

Apply the product rule to $59049 x^{5}$ .

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{59049^{\frac{1}{5}} {(x^{5})}^{\frac{1}{5}}}{9}$

Step 5.3.3.4

Rewrite $59049$ as $9^{5}$ .

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{{(9^{5})}^{\frac{1}{5}} {(x^{5})}^{\frac{1}{5}}}{9}$

Step 5.3.3.5

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

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{9^{5 (\frac{1}{5})} {(x^{5})}^{\frac{1}{5}}}{9}$

Step 5.3.3.6

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.3.3.6.1

Cancel the common factor.

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{9^{5 (\frac{1}{5})} {(x^{5})}^{\frac{1}{5}}}{9}$

Step 5.3.3.6.2

Rewrite the expression.

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{9 {(x^{5})}^{\frac{1}{5}}}{9}$

Step 5.3.3.7

Evaluate the exponent.

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{9 {(x^{5})}^{\frac{1}{5}}}{9}$

Step 5.3.3.8

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

Tap for more steps...

Step 5.3.3.8.1

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

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{9 x^{5 (\frac{1}{5})}}{9}$

Step 5.3.3.8.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.3.3.8.2.1

Cancel the common factor.

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{9 x^{5 (\frac{1}{5})}}{9}$

Step 5.3.3.8.2.2

Rewrite the expression.

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{9 x}{9}$

Step 5.3.3.9

Simplify.

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{9 x}{9}$

Step 5.3.4

Cancel the common factor of $9$ .

Tap for more steps...

Step 5.3.4.1

Cancel the common factor.

$f (\sqrt[7]{59049 x^{5} - 10}) = \frac{9 x}{9}$

Step 5.3.4.2

Divide $x$ by $1$ .

$f (\sqrt[7]{59049 x^{5} - 10}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \sqrt[7]{59049 x^{5} - 10}$ is the inverse of $f (x) = \frac{{(x^{7} + 10)}^{\frac{1}{5}}}{9}$ .

$f^{- 1} (x) = \sqrt[7]{59049 x^{5} - 10}$