Find the Inverse f(x)=((x^7+10)^(1/5))/9

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

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 $y$ .

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

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

Multiply both sides of the equation by $9$ .

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

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Simplify the exponent.

Simplify the left side.

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

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

Apply the product rule to $9 x$ .

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

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

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

Solve for $y$ .

Subtract $10$ from both sides of the equation.

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

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

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

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

Set up the composite result function.

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

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

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

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

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

Simplify the numerator.

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Reduce the expression by cancelling the common factors.

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

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

Cancel the common factor of $59049$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify by subtracting numbers.

Subtract $10$ from $10$ .

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

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

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

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

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

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

Set up the composite result function.

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

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

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

Simplify the numerator.

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

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

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

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

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

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

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

Cancel the common factor of $7$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

Add $- 10$ and $10$ .

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

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

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

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

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

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

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Cancel the common factor of $9$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

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

Find the Inverse f(x)=((x^7+10)^(1/5))/9

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