Algebra Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Divide each term in $5 {(\frac{y^{7}}{7})}^{\frac{1}{5}} = x$ by $5$ and simplify.

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Step 3.2.1

Divide each term in $5 {(\frac{y^{7}}{7})}^{\frac{1}{5}} = x$ by $5$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor.

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

Step 3.2.2.2

Simplify the expression.

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Step 3.2.2.2.1

Divide ${(\frac{y^{7}}{7})}^{\frac{1}{5}}$ by $1$ .

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

Step 3.2.2.2.2

Apply the product rule to $\frac{y^{7}}{7}$ .

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

Step 3.2.2.2.3

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

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Step 3.2.2.2.3.1

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

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

Step 3.2.2.2.3.2

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

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

Step 3.3

Multiply both sides of the equation by $7^{\frac{1}{5}}$ .

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

Step 3.4

Simplify both sides of the equation.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Cancel the common factor of $7^{\frac{1}{5}}$ .

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Step 3.4.1.1.1

Cancel the common factor.

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

Step 3.4.1.1.2

Rewrite the expression.

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

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $7^{\frac{1}{5}} \frac{x}{5}$ .

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Step 3.4.2.1.1

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

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

Step 3.4.2.1.2

Reorder factors in $\frac{7^{\frac{1}{5}} x}{5}$ .

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

Step 3.5

Raise each side of the equation to the power of $\frac{5}{7}$ to eliminate the fractional exponent on the left side.

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

Step 3.6

Simplify the exponent.

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Step 3.6.1

Simplify the left side.

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Step 3.6.1.1

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

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Step 3.6.1.1.1

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

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Step 3.6.1.1.1.1

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

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

Step 3.6.1.1.1.2

Cancel the common factor of $7$ .

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Step 3.6.1.1.1.2.1

Cancel the common factor.

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

Step 3.6.1.1.1.2.2

Rewrite the expression.

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

Step 3.6.1.1.1.3

Cancel the common factor of $5$ .

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Step 3.6.1.1.1.3.1

Cancel the common factor.

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

Step 3.6.1.1.1.3.2

Rewrite the expression.

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

Step 3.6.1.1.2

Simplify.

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

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify ${(\frac{x \cdot 7^{\frac{1}{5}}}{5})}^{\frac{5}{7}}$ .

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Step 3.6.2.1.1

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

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Step 3.6.2.1.1.1

Apply the product rule to $\frac{x \cdot 7^{\frac{1}{5}}}{5}$ .

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

Step 3.6.2.1.1.2

Apply the product rule to $x \cdot 7^{\frac{1}{5}}$ .

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

Step 3.6.2.1.2

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

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Step 3.6.2.1.2.1

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

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

Step 3.6.2.1.2.2

Cancel the common factor of $5$ .

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Step 3.6.2.1.2.2.1

Cancel the common factor.

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

Step 3.6.2.1.2.2.2

Rewrite the expression.

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

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{x^{\frac{5}{7}} \cdot 7^{\frac{1}{7}}}{5^{\frac{5}{7}}}$ is the inverse of $f (x) = 5 {(\frac{x^{7}}{7})}^{\frac{1}{5}}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

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

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Apply the product rule to $\frac{x^{7}}{7}$ .

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

Step 5.2.3.2

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

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Step 5.2.3.2.1

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

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

Step 5.2.3.2.2

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

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

Step 5.2.3.3

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

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

Step 5.2.3.4

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

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Step 5.2.3.4.1

Apply the product rule to $\frac{5 x^{\frac{7}{5}}}{7^{\frac{1}{5}}}$ .

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

Step 5.2.3.4.2

Apply the product rule to $5 x^{\frac{7}{5}}$ .

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

Step 5.2.3.5

Simplify the numerator.

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Step 5.2.3.5.1

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

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Step 5.2.3.5.1.1

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

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

Step 5.2.3.5.1.2

Cancel the common factor of $7$ .

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Step 5.2.3.5.1.2.1

Cancel the common factor.

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

Step 5.2.3.5.1.2.2

Rewrite the expression.

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

Step 5.2.3.5.1.3

Cancel the common factor of $5$ .

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Step 5.2.3.5.1.3.1

Cancel the common factor.

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

Step 5.2.3.5.1.3.2

Rewrite the expression.

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

Step 5.2.3.5.2

Simplify.

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

Step 5.2.3.6

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

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Step 5.2.3.6.1

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

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

Step 5.2.3.6.2

Cancel the common factor of $5$ .

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Step 5.2.3.6.2.1

Cancel the common factor.

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

Step 5.2.3.6.2.2

Rewrite the expression.

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

Step 5.2.4

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

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

Step 5.2.5

Reduce the expression by cancelling the common factors.

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Step 5.2.5.1

Reduce the expression $\frac{5^{\frac{5}{7}} x \cdot 7^{\frac{1}{7}}}{7^{\frac{1}{7}}}$ by cancelling the common factors.

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Step 5.2.5.1.1

Cancel the common factor.

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

Step 5.2.5.1.2

Rewrite the expression.

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

Step 5.2.5.2

Divide $5^{\frac{5}{7}} x$ by $1$ .

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

Step 5.2.6

Cancel the common factor.

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

Step 5.2.7

Divide $x$ by $1$ .

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

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

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

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

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Apply the product rule to $\frac{x^{\frac{5}{7}} \cdot 7^{\frac{1}{7}}}{5^{\frac{5}{7}}}$ .

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

Step 5.3.3.2

Simplify the numerator.

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Step 5.3.3.2.1

Apply the product rule to $x^{\frac{5}{7}} \cdot 7^{\frac{1}{7}}$ .

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

Step 5.3.3.2.2

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

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Step 5.3.3.2.2.1

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

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

Step 5.3.3.2.2.2

Cancel the common factor of $7$ .

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Step 5.3.3.2.2.2.1

Cancel the common factor.

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

Step 5.3.3.2.2.2.2

Rewrite the expression.

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

Step 5.3.3.2.3

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

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Step 5.3.3.2.3.1

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

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

Step 5.3.3.2.3.2

Cancel the common factor of $7$ .

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Step 5.3.3.2.3.2.1

Cancel the common factor.

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

Step 5.3.3.2.3.2.2

Rewrite the expression.

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

Step 5.3.3.2.4

Evaluate the exponent.

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

Step 5.3.3.3

Simplify the denominator.

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Step 5.3.3.3.1

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

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Step 5.3.3.3.1.1

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

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

Step 5.3.3.3.1.2

Cancel the common factor of $7$ .

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Step 5.3.3.3.1.2.1

Cancel the common factor.

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

Step 5.3.3.3.1.2.2

Rewrite the expression.

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

Step 5.3.3.3.2

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

$f (\frac{x^{\frac{5}{7}} \cdot 7^{\frac{1}{7}}}{5^{\frac{5}{7}}}) = 5 {(\frac{\frac{x^{5} \cdot 7}{3125}}{7})}^{\frac{1}{5}}$

Step 5.3.3.4

Move $7$ to the left of $x^{5}$ .

$f (\frac{x^{\frac{5}{7}} \cdot 7^{\frac{1}{7}}}{5^{\frac{5}{7}}}) = 5 {(\frac{\frac{7 x^{5}}{3125}}{7})}^{\frac{1}{5}}$

Step 5.3.4

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{x^{\frac{5}{7}} \cdot 7^{\frac{1}{7}}}{5^{\frac{5}{7}}}) = 5 {(\frac{7 x^{5}}{3125} \cdot \frac{1}{7})}^{\frac{1}{5}}$

Step 5.3.5

Simplify terms.

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Step 5.3.5.1

Combine.

$f (\frac{x^{\frac{5}{7}} \cdot 7^{\frac{1}{7}}}{5^{\frac{5}{7}}}) = 5 {(\frac{7 x^{5} \cdot 1}{3125 \cdot 7})}^{\frac{1}{5}}$

Step 5.3.5.2

Cancel the common factor of $7$ .

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Step 5.3.5.2.1

Cancel the common factor.

$f (\frac{x^{\frac{5}{7}} \cdot 7^{\frac{1}{7}}}{5^{\frac{5}{7}}}) = 5 {(\frac{7 x^{5} \cdot 1}{3125 \cdot 7})}^{\frac{1}{5}}$

Step 5.3.5.2.2

Rewrite the expression.

$f (\frac{x^{\frac{5}{7}} \cdot 7^{\frac{1}{7}}}{5^{\frac{5}{7}}}) = 5 {(\frac{x^{5} \cdot 1}{3125})}^{\frac{1}{5}}$

Step 5.3.5.3

Simplify the expression.

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Step 5.3.5.3.1

Multiply $x^{5}$ by $1$ .

$f (\frac{x^{\frac{5}{7}} \cdot 7^{\frac{1}{7}}}{5^{\frac{5}{7}}}) = 5 {(\frac{x^{5}}{3125})}^{\frac{1}{5}}$

Step 5.3.5.3.2

Apply the product rule to $\frac{x^{5}}{3125}$ .

$f (\frac{x^{\frac{5}{7}} \cdot 7^{\frac{1}{7}}}{5^{\frac{5}{7}}}) = 5 (\frac{{(x^{5})}^{\frac{1}{5}}}{3125^{\frac{1}{5}}})$

Step 5.3.6

Simplify the numerator.

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Step 5.3.6.1

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

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Step 5.3.6.1.1

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

$f (\frac{x^{\frac{5}{7}} \cdot 7^{\frac{1}{7}}}{5^{\frac{5}{7}}}) = 5 (\frac{x^{5 (\frac{1}{5})}}{3125^{\frac{1}{5}}})$

Step 5.3.6.1.2

Cancel the common factor of $5$ .

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Step 5.3.6.1.2.1

Cancel the common factor.

$f (\frac{x^{\frac{5}{7}} \cdot 7^{\frac{1}{7}}}{5^{\frac{5}{7}}}) = 5 (\frac{x^{5 (\frac{1}{5})}}{3125^{\frac{1}{5}}})$

Step 5.3.6.1.2.2

Rewrite the expression.

$f (\frac{x^{\frac{5}{7}} \cdot 7^{\frac{1}{7}}}{5^{\frac{5}{7}}}) = 5 (\frac{x}{3125^{\frac{1}{5}}})$

Step 5.3.6.2

Simplify.

$f (\frac{x^{\frac{5}{7}} \cdot 7^{\frac{1}{7}}}{5^{\frac{5}{7}}}) = 5 (\frac{x}{3125^{\frac{1}{5}}})$

Step 5.3.7

Simplify the denominator.

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Step 5.3.7.1

Rewrite $3125$ as $5^{5}$ .

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

Step 5.3.7.2

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

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

Step 5.3.7.3

Cancel the common factor of $5$ .

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Step 5.3.7.3.1

Cancel the common factor.

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

Step 5.3.7.3.2

Rewrite the expression.

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

Step 5.3.7.4

Evaluate the exponent.

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

Step 5.3.8

Cancel the common factor of $5$ .

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Step 5.3.8.1

Cancel the common factor.

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

Step 5.3.8.2

Rewrite the expression.

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

Step 5.4

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

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