Algebra Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Multiply both sides of the equation by $7$ .

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

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.

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

Step 3.3.1.2

Rewrite the expression.

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

Step 3.4

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

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

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 ${({(y^{3} + 7)}^{\frac{1}{5}})}^{5}$ .

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Step 3.5.1.1.1

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

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Step 3.5.1.1.1.1

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

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

Step 3.5.1.1.1.2

Cancel the common factor of $5$ .

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Step 3.5.1.1.1.2.1

Cancel the common factor.

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

Step 3.5.1.1.1.2.2

Rewrite the expression.

${(y^{3} + 7)}^{1} = {(7 x)}^{5}$

Step 3.5.1.1.2

Simplify.

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

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify ${(7 x)}^{5}$ .

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Step 3.5.2.1.1

Apply the product rule to $7 x$ .

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

Step 3.5.2.1.2

Raise $7$ to the power of $5$ .

$y^{3} + 7 = 16807 x^{5}$

Step 3.6

Solve for $y$ .

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Step 3.6.1

Subtract $7$ from both sides of the equation.

$y^{3} = 16807 x^{5} - 7$

Step 3.6.2

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

$y = \sqrt[3]{16807 x^{5} - 7}$

Step 3.6.3

Factor $7$ out of $16807 x^{5} - 7$ .

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Step 3.6.3.1

Factor $7$ out of $16807 x^{5}$ .

$y = \sqrt[3]{7 (2401 x^{5}) - 7}$

Step 3.6.3.2

Factor $7$ out of $- 7$ .

$y = \sqrt[3]{7 (2401 x^{5}) + 7 (- 1)}$

Step 3.6.3.3

Factor $7$ out of $7 (2401 x^{5}) + 7 (- 1)$ .

$y = \sqrt[3]{7 (2401 x^{5} - 1)}$

Step 4

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

$f^{- 1} (x) = \sqrt[3]{7 (2401 x^{5} - 1)}$

Step 5

Verify if $f^{- 1} (x) = \sqrt[3]{7 (2401 x^{5} - 1)}$ is the inverse of $f (x) = \frac{{(x^{3} + 7)}^{\frac{1}{5}}}{7}$ .

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

$f^{- 1} (\frac{{(x^{3} + 7)}^{\frac{1}{5}}}{7}) = \sqrt[3]{7 (2401 {(\frac{{(x^{3} + 7)}^{\frac{1}{5}}}{7})}^{5} - 1)}$

Step 5.2.3

Apply the product rule to $\frac{{(x^{3} + 7)}^{\frac{1}{5}}}{7}$ .

$f^{- 1} (\frac{{(x^{3} + 7)}^{\frac{1}{5}}}{7}) = \sqrt[3]{7 (2401 (\frac{{({(x^{3} + 7)}^{\frac{1}{5}})}^{5}}{7^{5}}) - 1)}$

Step 5.2.4

Simplify the numerator.

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Step 5.2.4.1

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

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Step 5.2.4.1.1

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

$f^{- 1} (\frac{{(x^{3} + 7)}^{\frac{1}{5}}}{7}) = \sqrt[3]{7 (2401 (\frac{{(x^{3} + 7)}^{\frac{1}{5} \cdot 5}}{7^{5}}) - 1)}$

Step 5.2.4.1.2

Cancel the common factor of $5$ .

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Step 5.2.4.1.2.1

Cancel the common factor.

$f^{- 1} (\frac{{(x^{3} + 7)}^{\frac{1}{5}}}{7}) = \sqrt[3]{7 (2401 (\frac{{(x^{3} + 7)}^{\frac{1}{5} \cdot 5}}{7^{5}}) - 1)}$

Step 5.2.4.1.2.2

Rewrite the expression.

$f^{- 1} (\frac{{(x^{3} + 7)}^{\frac{1}{5}}}{7}) = \sqrt[3]{7 (2401 (\frac{x^{3} + 7}{7^{5}}) - 1)}$

Step 5.2.4.2

Simplify.

$f^{- 1} (\frac{{(x^{3} + 7)}^{\frac{1}{5}}}{7}) = \sqrt[3]{7 (2401 (\frac{x^{3} + 7}{7^{5}}) - 1)}$

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 $5$ .

$f^{- 1} (\frac{{(x^{3} + 7)}^{\frac{1}{5}}}{7}) = \sqrt[3]{7 (2401 (\frac{x^{3} + 7}{16807}) - 1)}$

Step 5.2.5.2

Cancel the common factor of $2401$ .

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Step 5.2.5.2.1

Factor $2401$ out of $16807$ .

$f^{- 1} (\frac{{(x^{3} + 7)}^{\frac{1}{5}}}{7}) = \sqrt[3]{7 (2401 (\frac{x^{3} + 7}{2401 (7)}) - 1)}$

Step 5.2.5.2.2

Cancel the common factor.

$f^{- 1} (\frac{{(x^{3} + 7)}^{\frac{1}{5}}}{7}) = \sqrt[3]{7 (2401 (\frac{x^{3} + 7}{2401 \cdot 7}) - 1)}$

Step 5.2.5.2.3

Rewrite the expression.

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

Step 5.2.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

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

Step 5.2.7

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

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

Step 5.2.8

Combine the numerators over the common denominator.

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

Step 5.2.9

Rewrite $\frac{x^{3} + 7 - 1 \cdot 7}{7}$ in a factored form.

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Step 5.2.9.1

Multiply $- 1$ by $7$ .

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

Step 5.2.9.2

Subtract $7$ from $7$ .

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

Step 5.2.9.3

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

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

Step 5.2.10

Combine $7$ and $\frac{x^{3}}{7}$ .

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

Step 5.2.11

Reduce the expression by cancelling the common factors.

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Step 5.2.11.1

Reduce the expression $\frac{7 x^{3}}{7}$ by cancelling the common factors.

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Step 5.2.11.1.1

Cancel the common factor.

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

Step 5.2.11.1.2

Rewrite the expression.

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

Step 5.2.11.2

Divide $x^{3}$ by $1$ .

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

Step 5.2.12

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

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

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 $f (\sqrt[3]{7 (2401 x^{5} - 1)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{{({(\sqrt[3]{7 (2401 x^{5} - 1)})}^{3} + 7)}^{\frac{1}{5}}}{7}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Rewrite ${\sqrt[3]{7 (2401 x^{5} - 1)}}^{3}$ as $7 (2401 x^{5} - 1)$ .

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Step 5.3.3.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7 (2401 x^{5} - 1)}$ as ${(7 (2401 x^{5} - 1))}^{\frac{1}{3}}$ .

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{{({({(7 (2401 x^{5} - 1))}^{\frac{1}{3}})}^{3} + 7)}^{\frac{1}{5}}}{7}$

Step 5.3.3.1.1.2

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

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{{({(7 (2401 x^{5} - 1))}^{\frac{1}{3} \cdot 3} + 7)}^{\frac{1}{5}}}{7}$

Step 5.3.3.1.1.3

Combine $\frac{1}{3}$ and $3$ .

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{{({(7 (2401 x^{5} - 1))}^{\frac{3}{3}} + 7)}^{\frac{1}{5}}}{7}$

Step 5.3.3.1.1.4

Cancel the common factor of $3$ .

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Step 5.3.3.1.1.4.1

Cancel the common factor.

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{{({(7 (2401 x^{5} - 1))}^{\frac{3}{3}} + 7)}^{\frac{1}{5}}}{7}$

Step 5.3.3.1.1.4.2

Rewrite the expression.

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{{((7 (2401 x^{5} - 1)) + 7)}^{\frac{1}{5}}}{7}$

Step 5.3.3.1.1.5

Simplify.

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{{(7 (2401 x^{5} - 1) + 7)}^{\frac{1}{5}}}{7}$

Step 5.3.3.1.2

Apply the distributive property.

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{{(7 (2401 x^{5}) + 7 \cdot - 1 + 7)}^{\frac{1}{5}}}{7}$

Step 5.3.3.1.3

Multiply $2401$ by $7$ .

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{{(16807 x^{5} + 7 \cdot - 1 + 7)}^{\frac{1}{5}}}{7}$

Step 5.3.3.1.4

Multiply $7$ by $- 1$ .

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{{(16807 x^{5} - 7 + 7)}^{\frac{1}{5}}}{7}$

Step 5.3.3.2

Combine the opposite terms in $16807 x^{5} - 7 + 7$ .

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Step 5.3.3.2.1

Add $- 7$ and $7$ .

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{{(16807 x^{5} + 0)}^{\frac{1}{5}}}{7}$

Step 5.3.3.2.2

Add $16807 x^{5}$ and $0$ .

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{{(16807 x^{5})}^{\frac{1}{5}}}{7}$

Step 5.3.3.3

Apply the product rule to $16807 x^{5}$ .

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{16807^{\frac{1}{5}} {(x^{5})}^{\frac{1}{5}}}{7}$

Step 5.3.3.4

Rewrite $16807$ as $7^{5}$ .

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{{(7^{5})}^{\frac{1}{5}} {(x^{5})}^{\frac{1}{5}}}{7}$

Step 5.3.3.5

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

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{7^{5 (\frac{1}{5})} {(x^{5})}^{\frac{1}{5}}}{7}$

Step 5.3.3.6

Cancel the common factor of $5$ .

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Step 5.3.3.6.1

Cancel the common factor.

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{7^{5 (\frac{1}{5})} {(x^{5})}^{\frac{1}{5}}}{7}$

Step 5.3.3.6.2

Rewrite the expression.

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{7 {(x^{5})}^{\frac{1}{5}}}{7}$

Step 5.3.3.7

Evaluate the exponent.

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{7 {(x^{5})}^{\frac{1}{5}}}{7}$

Step 5.3.3.8

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

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Step 5.3.3.8.1

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

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{7 x^{5 (\frac{1}{5})}}{7}$

Step 5.3.3.8.2

Cancel the common factor of $5$ .

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Step 5.3.3.8.2.1

Cancel the common factor.

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{7 x^{5 (\frac{1}{5})}}{7}$

Step 5.3.3.8.2.2

Rewrite the expression.

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{7 x}{7}$

Step 5.3.3.9

Simplify.

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{7 x}{7}$

Step 5.3.4

Cancel the common factor of $7$ .

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Step 5.3.4.1

Cancel the common factor.

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = \frac{7 x}{7}$

Step 5.3.4.2

Divide $x$ by $1$ .

$f (\sqrt[3]{7 (2401 x^{5} - 1)}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \sqrt[3]{7 (2401 x^{5} - 1)}$ is the inverse of $f (x) = \frac{{(x^{3} + 7)}^{\frac{1}{5}}}{7}$ .

$f^{- 1} (x) = \sqrt[3]{7 (2401 x^{5} - 1)}$