Algebra Examples

$f (x) = {(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}$

Step 1

Write $f (x) = {(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}$ as an equation.

$y = {(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}$

Step 2

Interchange the variables.

$x = {(\frac{y^{7} - 2}{3})}^{\frac{1}{5}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as ${(\frac{y^{7} - 2}{3})}^{\frac{1}{5}} = x$ .

${(\frac{y^{7} - 2}{3})}^{\frac{1}{5}} = x$

Step 3.2

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

${({(\frac{y^{7} - 2}{3})}^{\frac{1}{5}})}^{5} = x^{5}$

Step 3.3

Simplify the left side.

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Step 3.3.1

Simplify ${({(\frac{y^{7} - 2}{3})}^{\frac{1}{5}})}^{5}$ .

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Step 3.3.1.1

Multiply the exponents in ${({(\frac{y^{7} - 2}{3})}^{\frac{1}{5}})}^{5}$ .

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Step 3.3.1.1.1

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

${(\frac{y^{7} - 2}{3})}^{\frac{1}{5} \cdot 5} = x^{5}$

Step 3.3.1.1.2

Cancel the common factor of $5$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

${(\frac{y^{7} - 2}{3})}^{\frac{1}{5} \cdot 5} = x^{5}$

Step 3.3.1.1.2.2

Rewrite the expression.

${(\frac{y^{7} - 2}{3})}^{1} = x^{5}$

Step 3.3.1.2

Split the fraction $\frac{y^{7} - 2}{3}$ into two fractions.

${(\frac{y^{7}}{3} + \frac{- 2}{3})}^{1} = x^{5}$

Step 3.3.1.3

Move the negative in front of the fraction.

${(\frac{y^{7}}{3} - \frac{2}{3})}^{1} = x^{5}$

Step 3.3.1.4

Simplify.

$\frac{y^{7}}{3} - \frac{2}{3} = x^{5}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Add $\frac{2}{3}$ to both sides of the equation.

$\frac{y^{7}}{3} = x^{5} + \frac{2}{3}$

Step 3.4.2

Multiply both sides of the equation by $3$ .

$3 \frac{y^{7}}{3} = 3 (x^{5} + \frac{2}{3})$

Step 3.4.3

Simplify both sides of the equation.

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Step 3.4.3.1

Simplify the left side.

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Step 3.4.3.1.1

Cancel the common factor of $3$ .

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Step 3.4.3.1.1.1

Cancel the common factor.

$3 \frac{y^{7}}{3} = 3 (x^{5} + \frac{2}{3})$

Step 3.4.3.1.1.2

Rewrite the expression.

$y^{7} = 3 (x^{5} + \frac{2}{3})$

Step 3.4.3.2

Simplify the right side.

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Step 3.4.3.2.1

Simplify $3 (x^{5} + \frac{2}{3})$ .

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Step 3.4.3.2.1.1

Apply the distributive property.

$y^{7} = 3 x^{5} + 3 (\frac{2}{3})$

Step 3.4.3.2.1.2

Cancel the common factor of $3$ .

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Step 3.4.3.2.1.2.1

Cancel the common factor.

$y^{7} = 3 x^{5} + 3 (\frac{2}{3})$

Step 3.4.3.2.1.2.2

Rewrite the expression.

$y^{7} = 3 x^{5} + 2$

Step 3.4.4

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

$y = \sqrt[7]{3 x^{5} + 2}$

Step 4

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

$f^{- 1} (x) = \sqrt[7]{3 x^{5} + 2}$

Step 5

Verify if $f^{- 1} (x) = \sqrt[7]{3 x^{5} + 2}$ is the inverse of $f (x) = {(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}$ .

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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 $f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{3 {({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}})}^{5} + 2}$

Step 5.2.3

Multiply the exponents in ${({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}})}^{5}$ .

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Step 5.2.3.1

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

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{3 {(\frac{x^{7} - 2}{3})}^{\frac{1}{5} \cdot 5} + 2}$

Step 5.2.3.2

Cancel the common factor of $5$ .

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Step 5.2.3.2.1

Cancel the common factor.

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{3 {(\frac{x^{7} - 2}{3})}^{\frac{1}{5} \cdot 5} + 2}$

Step 5.2.3.2.2

Rewrite the expression.

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{3 (\frac{x^{7} - 2}{3}) + 2}$

Step 5.2.4

Cancel the common factor of $3$ .

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Step 5.2.4.1

Cancel the common factor.

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{3 (\frac{x^{7} - 2}{3}) + 2}$

Step 5.2.4.2

Rewrite the expression.

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{x^{7} - 2 + 2}$

Step 5.2.5

Simplify by adding numbers.

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Step 5.2.5.1

Add $- 2$ and $2$ .

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{x^{7} + 0}$

Step 5.2.5.2

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

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{x^{7}}$

Step 5.2.6

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

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = x$

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 $f (\sqrt[7]{3 x^{5} + 2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{{(\sqrt[7]{3 x^{5} + 2})}^{7} - 2}{3})}^{\frac{1}{5}}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{3 x^{5} + 2}$ as ${(3 x^{5} + 2)}^{\frac{1}{7}}$ .

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{{({(3 x^{5} + 2)}^{\frac{1}{7}})}^{7} - 2}{3})}^{\frac{1}{5}}$

Step 5.3.3.2

Multiply the exponents in ${({(3 x^{5} + 2)}^{\frac{1}{7}})}^{7}$ .

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Step 5.3.3.2.1

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

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{{(3 x^{5} + 2)}^{\frac{1}{7} \cdot 7} - 2}{3})}^{\frac{1}{5}}$

Step 5.3.3.2.2

Cancel the common factor of $7$ .

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Step 5.3.3.2.2.1

Cancel the common factor.

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{{(3 x^{5} + 2)}^{\frac{1}{7} \cdot 7} - 2}{3})}^{\frac{1}{5}}$

Step 5.3.3.2.2.2

Rewrite the expression.

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{(3 x^{5} + 2) - 2}{3})}^{\frac{1}{5}}$

Step 5.3.3.3

Simplify.

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{3 x^{5} + 2 - 2}{3})}^{\frac{1}{5}}$

Step 5.3.3.4

Subtract $2$ from $2$ .

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{3 x^{5} + 0}{3})}^{\frac{1}{5}}$

Step 5.3.3.5

Add $3 x^{5}$ and $0$ .

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{3 x^{5}}{3})}^{\frac{1}{5}}$

Step 5.3.4

Reduce the expression by cancelling the common factors.

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Step 5.3.4.1

Cancel the common factor of $3$ .

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Step 5.3.4.1.1

Cancel the common factor.

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{3 x^{5}}{3})}^{\frac{1}{5}}$

Step 5.3.4.1.2

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

$f (\sqrt[7]{3 x^{5} + 2}) = {(x^{5})}^{\frac{1}{5}}$

Step 5.3.4.2

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

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Step 5.3.4.2.1

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

$f (\sqrt[7]{3 x^{5} + 2}) = x^{5 (\frac{1}{5})}$

Step 5.3.4.2.2

Cancel the common factor of $5$ .

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Step 5.3.4.2.2.1

Cancel the common factor.

$f (\sqrt[7]{3 x^{5} + 2}) = x^{5 (\frac{1}{5})}$

Step 5.3.4.2.2.2

Rewrite the expression.

$f (\sqrt[7]{3 x^{5} + 2}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \sqrt[7]{3 x^{5} + 2}$ is the inverse of $f (x) = {(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}$ .

$f^{- 1} (x) = \sqrt[7]{3 x^{5} + 2}$