Algebra Examples

$\frac{{(x + 3)}^{\frac{1}{3}}}{8}$

Step 1

Interchange the variables.

$x = \frac{{(y + 3)}^{\frac{1}{3}}}{8}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{{(y + 3)}^{\frac{1}{3}}}{8} = x$ .

$\frac{{(y + 3)}^{\frac{1}{3}}}{8} = x$

Step 2.2

Multiply both sides of the equation by $8$ .

$8 \frac{{(y + 3)}^{\frac{1}{3}}}{8} = 8 x$

Step 2.3

Simplify the left side.

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Step 2.3.1

Cancel the common factor of $8$ .

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Step 2.3.1.1

Cancel the common factor.

$8 \frac{{(y + 3)}^{\frac{1}{3}}}{8} = 8 x$

Step 2.3.1.2

Rewrite the expression.

${(y + 3)}^{\frac{1}{3}} = 8 x$

Step 2.4

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${({(y + 3)}^{\frac{1}{3}})}^{3} = {(8 x)}^{3}$

Step 2.5

Simplify the exponent.

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Step 2.5.1

Simplify the left side.

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Step 2.5.1.1

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

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Step 2.5.1.1.1

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

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Step 2.5.1.1.1.1

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

${(y + 3)}^{\frac{1}{3} \cdot 3} = {(8 x)}^{3}$

Step 2.5.1.1.1.2

Cancel the common factor of $3$ .

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Step 2.5.1.1.1.2.1

Cancel the common factor.

${(y + 3)}^{\frac{1}{3} \cdot 3} = {(8 x)}^{3}$

Step 2.5.1.1.1.2.2

Rewrite the expression.

${(y + 3)}^{1} = {(8 x)}^{3}$

Step 2.5.1.1.2

Simplify.

$y + 3 = {(8 x)}^{3}$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify ${(8 x)}^{3}$ .

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Step 2.5.2.1.1

Apply the product rule to $8 x$ .

$y + 3 = 8^{3} x^{3}$

Step 2.5.2.1.2

Raise $8$ to the power of $3$ .

$y + 3 = 512 x^{3}$

Step 2.6

Subtract $3$ from both sides of the equation.

$y = 512 x^{3} - 3$

Step 3

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

$f^{- 1} (x) = 512 x^{3} - 3$

Step 4

Verify if $f^{- 1} (x) = 512 x^{3} - 3$ is the inverse of $f (x) = \frac{{(x + 3)}^{\frac{1}{3}}}{8}$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (\frac{{(x + 3)}^{\frac{1}{3}}}{8})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{{(x + 3)}^{\frac{1}{3}}}{8}) = 512 {(\frac{{(x + 3)}^{\frac{1}{3}}}{8})}^{3} - 3$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Apply the product rule to $\frac{{(x + 3)}^{\frac{1}{3}}}{8}$ .

$f^{- 1} (\frac{{(x + 3)}^{\frac{1}{3}}}{8}) = 512 (\frac{{({(x + 3)}^{\frac{1}{3}})}^{3}}{8^{3}}) - 3$

Step 4.2.3.2

Simplify the numerator.

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Step 4.2.3.2.1

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

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Step 4.2.3.2.1.1

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

$f^{- 1} (\frac{{(x + 3)}^{\frac{1}{3}}}{8}) = 512 (\frac{{(x + 3)}^{\frac{1}{3} \cdot 3}}{8^{3}}) - 3$

Step 4.2.3.2.1.2

Cancel the common factor of $3$ .

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Step 4.2.3.2.1.2.1

Cancel the common factor.

$f^{- 1} (\frac{{(x + 3)}^{\frac{1}{3}}}{8}) = 512 (\frac{{(x + 3)}^{\frac{1}{3} \cdot 3}}{8^{3}}) - 3$

Step 4.2.3.2.1.2.2

Rewrite the expression.

$f^{- 1} (\frac{{(x + 3)}^{\frac{1}{3}}}{8}) = 512 (\frac{x + 3}{8^{3}}) - 3$

Step 4.2.3.2.2

Simplify.

$f^{- 1} (\frac{{(x + 3)}^{\frac{1}{3}}}{8}) = 512 (\frac{x + 3}{8^{3}}) - 3$

Step 4.2.3.3

Raise $8$ to the power of $3$ .

$f^{- 1} (\frac{{(x + 3)}^{\frac{1}{3}}}{8}) = 512 (\frac{x + 3}{512}) - 3$

Step 4.2.3.4

Cancel the common factor of $512$ .

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Step 4.2.3.4.1

Cancel the common factor.

$f^{- 1} (\frac{{(x + 3)}^{\frac{1}{3}}}{8}) = 512 (\frac{x + 3}{512}) - 3$

Step 4.2.3.4.2

Rewrite the expression.

$f^{- 1} (\frac{{(x + 3)}^{\frac{1}{3}}}{8}) = x + 3 - 3$

Step 4.2.4

Combine the opposite terms in $x + 3 - 3$ .

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Step 4.2.4.1

Subtract $3$ from $3$ .

$f^{- 1} (\frac{{(x + 3)}^{\frac{1}{3}}}{8}) = x + 0$

Step 4.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\frac{{(x + 3)}^{\frac{1}{3}}}{8}) = x$

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (512 x^{3} - 3)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (512 x^{3} - 3) = \frac{{((512 x^{3} - 3) + 3)}^{\frac{1}{3}}}{8}$

Step 4.3.3

Simplify the numerator.

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Step 4.3.3.1

Combine the opposite terms in $512 x^{3} - 3 + 3$ .

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Step 4.3.3.1.1

Add $- 3$ and $3$ .

$f (512 x^{3} - 3) = \frac{{(512 x^{3} + 0)}^{\frac{1}{3}}}{8}$

Step 4.3.3.1.2

Add $512 x^{3}$ and $0$ .

$f (512 x^{3} - 3) = \frac{{(512 x^{3})}^{\frac{1}{3}}}{8}$

Step 4.3.3.2

Apply the product rule to $512 x^{3}$ .

$f (512 x^{3} - 3) = \frac{512^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}}}{8}$

Step 4.3.3.3

Rewrite $512$ as $8^{3}$ .

$f (512 x^{3} - 3) = \frac{{(8^{3})}^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}}}{8}$

Step 4.3.3.4

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

$f (512 x^{3} - 3) = \frac{8^{3 (\frac{1}{3})} {(x^{3})}^{\frac{1}{3}}}{8}$

Step 4.3.3.5

Cancel the common factor of $3$ .

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Step 4.3.3.5.1

Cancel the common factor.

$f (512 x^{3} - 3) = \frac{8^{3 (\frac{1}{3})} {(x^{3})}^{\frac{1}{3}}}{8}$

Step 4.3.3.5.2

Rewrite the expression.

$f (512 x^{3} - 3) = \frac{8 {(x^{3})}^{\frac{1}{3}}}{8}$

Step 4.3.3.6

Evaluate the exponent.

$f (512 x^{3} - 3) = \frac{8 {(x^{3})}^{\frac{1}{3}}}{8}$

Step 4.3.3.7

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

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Step 4.3.3.7.1

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

$f (512 x^{3} - 3) = \frac{8 x^{3 (\frac{1}{3})}}{8}$

Step 4.3.3.7.2

Cancel the common factor of $3$ .

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Step 4.3.3.7.2.1

Cancel the common factor.

$f (512 x^{3} - 3) = \frac{8 x^{3 (\frac{1}{3})}}{8}$

Step 4.3.3.7.2.2

Rewrite the expression.

$f (512 x^{3} - 3) = \frac{8 x}{8}$

Step 4.3.3.8

Simplify.

$f (512 x^{3} - 3) = \frac{8 x}{8}$

Step 4.3.4

Cancel the common factor of $8$ .

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Step 4.3.4.1

Cancel the common factor.

$f (512 x^{3} - 3) = \frac{8 x}{8}$

Step 4.3.4.2

Divide $x$ by $1$ .

$f (512 x^{3} - 3) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 512 x^{3} - 3$ is the inverse of $f (x) = \frac{{(x + 3)}^{\frac{1}{3}}}{8}$ .

$f^{- 1} (x) = 512 x^{3} - 3$