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

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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

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

Raise each side of the equation to the power of $7$ to eliminate the fractional exponent on the left side.

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $7$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$\frac{y^{5}}{10} = x^{7}$

Solve for $y$ .

Multiply both sides of the equation by $10$ .

$10 \frac{y^{5}}{10} = 10 x^{7}$

Simplify the left side.

Cancel the common factor of $10$ .

Cancel the common factor.

$10 \frac{y^{5}}{10} = 10 x^{7}$

Rewrite the expression.

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

Take the 5th root of both sides of the equation to eliminate the exponent on the left side.

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

Simplify $\sqrt[5]{10 x^{7}}$ .

Rewrite $10 x^{7}$ as $x^{5} \cdot (10 x^{2})$ .

Factor out $x^{5}$ .

$y = \sqrt[5]{10 (x^{5} x^{2})}$

Reorder $10$ and $x^{5}$ .

$y = \sqrt[5]{x^{5} \cdot 10 x^{2}}$

Add parentheses.

$y = \sqrt[5]{x^{5} \cdot (10 x^{2})}$

Pull terms out from under the radical.

$y = x \sqrt[5]{10 x^{2}}$

Step 4

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

$f^{- 1} (x) = x \sqrt[5]{10 x^{2}}$

Step 5

Verify if $f^{- 1} (x) = x \sqrt[5]{10 x^{2}}$ is the inverse of $f (x) = {(\frac{x^{5}}{10})}^{\frac{1}{7}}$ .

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

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) \sqrt[5]{10 {({(\frac{x^{5}}{10})}^{\frac{1}{7}})}^{2}}$

Apply basic rules of exponents.

Apply the product rule to $\frac{x^{5}}{10}$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{{(x^{5})}^{\frac{1}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 {({(\frac{x^{5}}{10})}^{\frac{1}{7}})}^{2}}$

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

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

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{5 (\frac{1}{7})}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 {({(\frac{x^{5}}{10})}^{\frac{1}{7}})}^{2}}$

Combine $5$ and $\frac{1}{7}$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 {({(\frac{x^{5}}{10})}^{\frac{1}{7}})}^{2}}$

Multiply the exponents in ${({(\frac{x^{5}}{10})}^{\frac{1}{7}})}^{2}$ .

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

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 {(\frac{x^{5}}{10})}^{\frac{1}{7} \cdot 2}}$

Combine $\frac{1}{7}$ and $2$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 {(\frac{x^{5}}{10})}^{\frac{2}{7}}}$

Apply the product rule to $\frac{x^{5}}{10}$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 (\frac{{(x^{5})}^{\frac{2}{7}}}{10^{\frac{2}{7}}})}$

Multiply the exponents in ${(x^{5})}^{\frac{2}{7}}$ .

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

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 (\frac{x^{5 (\frac{2}{7})}}{10^{\frac{2}{7}}})}$

Multiply $5 (\frac{2}{7})$ .

Combine $5$ and $\frac{2}{7}$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 (\frac{x^{\frac{5 \cdot 2}{7}}}{10^{\frac{2}{7}}})}$

Multiply $5$ by $2$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 (\frac{x^{\frac{10}{7}}}{10^{\frac{2}{7}}})}$

Combine $10$ and $\frac{x^{\frac{10}{7}}}{10^{\frac{2}{7}}}$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{\frac{10 x^{\frac{10}{7}}}{10^{\frac{2}{7}}}}$

Reduce the expression by cancelling the common factors.

Move $10^{\frac{2}{7}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 x^{\frac{10}{7}} \cdot 10^{- \frac{2}{7}}}$

Multiply $10$ by $10^{- \frac{2}{7}}$ by adding the exponents.

Move $10^{- \frac{2}{7}}$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10^{- \frac{2}{7}} \cdot (10 x^{\frac{10}{7}})}$

Multiply $10^{- \frac{2}{7}}$ by $10$ .

Raise $10$ to the power of $1$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10^{- \frac{2}{7}} \cdot (10 x^{\frac{10}{7}})}$

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

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10^{- \frac{2}{7} + 1} x^{\frac{10}{7}}}$

Write $1$ as a fraction with a common denominator.

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10^{- \frac{2}{7} + \frac{7}{7}} x^{\frac{10}{7}}}$

Combine the numerators over the common denominator.

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10^{\frac{- 2 + 7}{7}} x^{\frac{10}{7}}}$

Add $- 2$ and $7$ .

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

Rewrite $10^{\frac{5}{7}} x^{\frac{10}{7}}$ as ${(10^{\frac{1}{7}} x^{\frac{2}{7}})}^{5}$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{{(10^{\frac{1}{7}} x^{\frac{2}{7}})}^{5}}$

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

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot (10^{\frac{1}{7}} x^{\frac{2}{7}})$

Rewrite using the commutative property of multiplication.

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = 10^{\frac{1}{7}} (\frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}}) \cdot x^{\frac{2}{7}}$

Cancel the common factor of $10^{\frac{1}{7}}$ .

Cancel the common factor.

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = 10^{\frac{1}{7}} (\frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}}) \cdot x^{\frac{2}{7}}$

Rewrite the expression.

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = x^{\frac{5}{7}} x^{\frac{2}{7}}$

Multiply $x^{\frac{5}{7}}$ by $x^{\frac{2}{7}}$ by adding the exponents.

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

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = x^{\frac{5}{7} + \frac{2}{7}}$

Combine the numerators over the common denominator.

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = x^{\frac{5 + 2}{7}}$

Add $5$ and $2$ .

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

Divide $7$ by $7$ .

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

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

Set up the composite result function.

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

Evaluate $f (x \sqrt[5]{10 x^{2}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{{(x \sqrt[5]{10 x^{2}})}^{5}}{10})}^{\frac{1}{7}}$

Simplify the numerator.

Apply the product rule to $x \sqrt[5]{10 x^{2}}$ .

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{5} {\sqrt[5]{10 x^{2}}}^{5}}{10})}^{\frac{1}{7}}$

Rewrite ${\sqrt[5]{10 x^{2}}}^{5}$ as $10 x^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{10 x^{2}}$ as ${(10 x^{2})}^{\frac{1}{5}}$ .

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{5} {({(10 x^{2})}^{\frac{1}{5}})}^{5}}{10})}^{\frac{1}{7}}$

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

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{5} {(10 x^{2})}^{\frac{1}{5} \cdot 5}}{10})}^{\frac{1}{7}}$

Combine $\frac{1}{5}$ and $5$ .

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{5} {(10 x^{2})}^{\frac{5}{5}}}{10})}^{\frac{1}{7}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{5} {(10 x^{2})}^{\frac{5}{5}}}{10})}^{\frac{1}{7}}$

Rewrite the expression.

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{5} (10 x^{2})}{10})}^{\frac{1}{7}}$

Simplify.

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{5} (10 x^{2})}{10})}^{\frac{1}{7}}$

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{5} \cdot (10 x^{2})}{10})}^{\frac{1}{7}}$

Multiply $x^{5}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{2} x^{5} \cdot 10}{10})}^{\frac{1}{7}}$

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

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{2 + 5} \cdot 10}{10})}^{\frac{1}{7}}$

Add $2$ and $5$ .

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{7} \cdot 10}{10})}^{\frac{1}{7}}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $10$ .

Cancel the common factor.

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{7} \cdot 10}{10})}^{\frac{1}{7}}$

Divide $x^{7}$ by $1$ .

$f (x \sqrt[5]{10 x^{2}}) = {(x^{7})}^{\frac{1}{7}}$

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

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

$f (x \sqrt[5]{10 x^{2}}) = x^{7 (\frac{1}{7})}$

Cancel the common factor of $7$ .

Cancel the common factor.

$f (x \sqrt[5]{10 x^{2}}) = x^{7 (\frac{1}{7})}$

Rewrite the expression.

$f (x \sqrt[5]{10 x^{2}}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x \sqrt[5]{10 x^{2}}$ is the inverse of $f (x) = {(\frac{x^{5}}{10})}^{\frac{1}{7}}$ .

$f^{- 1} (x) = x \sqrt[5]{10 x^{2}}$

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

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