Algebra Examples

$f (x) = \frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7}$

Step 1

Write $f (x) = \frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7}$ as an equation.

$y = \frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7}$

Step 2

Interchange the variables.

$x = \frac{{(y^{5} - 8)}^{\frac{1}{7}}}{7}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{{(y^{5} - 8)}^{\frac{1}{7}}}{7} = x$ .

$\frac{{(y^{5} - 8)}^{\frac{1}{7}}}{7} = x$

Step 3.2

Multiply both sides of the equation by $7$ .

$7 \frac{{(y^{5} - 8)}^{\frac{1}{7}}}{7} = 7 x$

Step 3.3

Simplify the left side.

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Step 3.3.1

Cancel the common factor of $7$ .

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Step 3.3.1.1

Cancel the common factor.

$7 \frac{{(y^{5} - 8)}^{\frac{1}{7}}}{7} = 7 x$

Step 3.3.1.2

Rewrite the expression.

${(y^{5} - 8)}^{\frac{1}{7}} = 7 x$

Step 3.4

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

${({(y^{5} - 8)}^{\frac{1}{7}})}^{7} = {(7 x)}^{7}$

Step 3.5

Simplify the exponent.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

Simplify ${({(y^{5} - 8)}^{\frac{1}{7}})}^{7}$ .

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Step 3.5.1.1.1

Multiply the exponents in ${({(y^{5} - 8)}^{\frac{1}{7}})}^{7}$ .

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Step 3.5.1.1.1.1

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

${(y^{5} - 8)}^{\frac{1}{7} \cdot 7} = {(7 x)}^{7}$

Step 3.5.1.1.1.2

Cancel the common factor of $7$ .

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Step 3.5.1.1.1.2.1

Cancel the common factor.

${(y^{5} - 8)}^{\frac{1}{7} \cdot 7} = {(7 x)}^{7}$

Step 3.5.1.1.1.2.2

Rewrite the expression.

${(y^{5} - 8)}^{1} = {(7 x)}^{7}$

Step 3.5.1.1.2

Simplify.

$y^{5} - 8 = {(7 x)}^{7}$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify ${(7 x)}^{7}$ .

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Step 3.5.2.1.1

Apply the product rule to $7 x$ .

$y^{5} - 8 = 7^{7} x^{7}$

Step 3.5.2.1.2

Raise $7$ to the power of $7$ .

$y^{5} - 8 = 823543 x^{7}$

Step 3.6

Solve for $y$ .

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Step 3.6.1

Add $8$ to both sides of the equation.

$y^{5} = 823543 x^{7} + 8$

Step 3.6.2

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

$y = \sqrt[5]{823543 x^{7} + 8}$

Step 4

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

$f^{- 1} (x) = \sqrt[5]{823543 x^{7} + 8}$

Step 5

Verify if $f^{- 1} (x) = \sqrt[5]{823543 x^{7} + 8}$ is the inverse of $f (x) = \frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7}$ .

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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 $f^{- 1} (\frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7}) = \sqrt[5]{823543 {(\frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7})}^{7} + 8}$

Step 5.2.3

Apply the product rule to $\frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7}$ .

$f^{- 1} (\frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7}) = \sqrt[5]{823543 (\frac{{({(x^{5} - 8)}^{\frac{1}{7}})}^{7}}{7^{7}}) + 8}$

Step 5.2.4

Simplify the numerator.

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Step 5.2.4.1

Multiply the exponents in ${({(x^{5} - 8)}^{\frac{1}{7}})}^{7}$ .

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Step 5.2.4.1.1

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

$f^{- 1} (\frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7}) = \sqrt[5]{823543 (\frac{{(x^{5} - 8)}^{\frac{1}{7} \cdot 7}}{7^{7}}) + 8}$

Step 5.2.4.1.2

Cancel the common factor of $7$ .

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Step 5.2.4.1.2.1

Cancel the common factor.

$f^{- 1} (\frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7}) = \sqrt[5]{823543 (\frac{{(x^{5} - 8)}^{\frac{1}{7} \cdot 7}}{7^{7}}) + 8}$

Step 5.2.4.1.2.2

Rewrite the expression.

$f^{- 1} (\frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7}) = \sqrt[5]{823543 (\frac{x^{5} - 8}{7^{7}}) + 8}$

Step 5.2.4.2

Simplify.

$f^{- 1} (\frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7}) = \sqrt[5]{823543 (\frac{x^{5} - 8}{7^{7}}) + 8}$

Step 5.2.5

Reduce the expression by cancelling the common factors.

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Step 5.2.5.1

Raise $7$ to the power of $7$ .

$f^{- 1} (\frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7}) = \sqrt[5]{823543 (\frac{x^{5} - 8}{823543}) + 8}$

Step 5.2.5.2

Cancel the common factor of $823543$ .

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Step 5.2.5.2.1

Cancel the common factor.

$f^{- 1} (\frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7}) = \sqrt[5]{823543 (\frac{x^{5} - 8}{823543}) + 8}$

Step 5.2.5.2.2

Rewrite the expression.

$f^{- 1} (\frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7}) = \sqrt[5]{x^{5} - 8 + 8}$

Step 5.2.5.3

Simplify by adding numbers.

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Step 5.2.5.3.1

Add $- 8$ and $8$ .

$f^{- 1} (\frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7}) = \sqrt[5]{x^{5} + 0}$

Step 5.2.5.3.2

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

$f^{- 1} (\frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7}) = \sqrt[5]{x^{5}}$

Step 5.2.6

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

$f^{- 1} (\frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7}) = x$

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 $f (\sqrt[5]{823543 x^{7} + 8})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{{({(\sqrt[5]{823543 x^{7} + 8})}^{5} - 8)}^{\frac{1}{7}}}{7}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Rewrite ${\sqrt[5]{823543 x^{7} + 8}}^{5}$ as $823543 x^{7} + 8$ .

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Step 5.3.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{823543 x^{7} + 8}$ as ${(823543 x^{7} + 8)}^{\frac{1}{5}}$ .

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{{({({(823543 x^{7} + 8)}^{\frac{1}{5}})}^{5} - 8)}^{\frac{1}{7}}}{7}$

Step 5.3.3.1.2

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

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{{({(823543 x^{7} + 8)}^{\frac{1}{5} \cdot 5} - 8)}^{\frac{1}{7}}}{7}$

Step 5.3.3.1.3

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

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{{({(823543 x^{7} + 8)}^{\frac{5}{5}} - 8)}^{\frac{1}{7}}}{7}$

Step 5.3.3.1.4

Cancel the common factor of $5$ .

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Step 5.3.3.1.4.1

Cancel the common factor.

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{{({(823543 x^{7} + 8)}^{\frac{5}{5}} - 8)}^{\frac{1}{7}}}{7}$

Step 5.3.3.1.4.2

Rewrite the expression.

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{{((823543 x^{7} + 8) - 8)}^{\frac{1}{7}}}{7}$

Step 5.3.3.1.5

Simplify.

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{{(823543 x^{7} + 8 - 8)}^{\frac{1}{7}}}{7}$

Step 5.3.3.2

Combine the opposite terms in $823543 x^{7} + 8 - 8$ .

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Step 5.3.3.2.1

Subtract $8$ from $8$ .

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{{(823543 x^{7} + 0)}^{\frac{1}{7}}}{7}$

Step 5.3.3.2.2

Add $823543 x^{7}$ and $0$ .

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{{(823543 x^{7})}^{\frac{1}{7}}}{7}$

Step 5.3.3.3

Apply the product rule to $823543 x^{7}$ .

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{823543^{\frac{1}{7}} {(x^{7})}^{\frac{1}{7}}}{7}$

Step 5.3.3.4

Rewrite $823543$ as $7^{7}$ .

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{{(7^{7})}^{\frac{1}{7}} {(x^{7})}^{\frac{1}{7}}}{7}$

Step 5.3.3.5

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

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{7^{7 (\frac{1}{7})} {(x^{7})}^{\frac{1}{7}}}{7}$

Step 5.3.3.6

Cancel the common factor of $7$ .

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Step 5.3.3.6.1

Cancel the common factor.

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{7^{7 (\frac{1}{7})} {(x^{7})}^{\frac{1}{7}}}{7}$

Step 5.3.3.6.2

Rewrite the expression.

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{7 {(x^{7})}^{\frac{1}{7}}}{7}$

Step 5.3.3.7

Evaluate the exponent.

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{7 {(x^{7})}^{\frac{1}{7}}}{7}$

Step 5.3.3.8

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

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Step 5.3.3.8.1

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

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{7 x^{7 (\frac{1}{7})}}{7}$

Step 5.3.3.8.2

Cancel the common factor of $7$ .

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Step 5.3.3.8.2.1

Cancel the common factor.

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{7 x^{7 (\frac{1}{7})}}{7}$

Step 5.3.3.8.2.2

Rewrite the expression.

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{7 x}{7}$

Step 5.3.3.9

Simplify.

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{7 x}{7}$

Step 5.3.4

Cancel the common factor of $7$ .

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Step 5.3.4.1

Cancel the common factor.

$f (\sqrt[5]{823543 x^{7} + 8}) = \frac{7 x}{7}$

Step 5.3.4.2

Divide $x$ by $1$ .

$f (\sqrt[5]{823543 x^{7} + 8}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \sqrt[5]{823543 x^{7} + 8}$ is the inverse of $f (x) = \frac{{(x^{5} - 8)}^{\frac{1}{7}}}{7}$ .

$f^{- 1} (x) = \sqrt[5]{823543 x^{7} + 8}$