Algebra Examples

$10 x^{\frac{1}{3}}$

Step 1

Interchange the variables.

$x = 10 y^{\frac{1}{3}}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $10 y^{\frac{1}{3}} = x$ .

$10 y^{\frac{1}{3}} = x$

Step 2.2

Divide each term in $10 y^{\frac{1}{3}} = x$ by $10$ and simplify.

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Step 2.2.1

Divide each term in $10 y^{\frac{1}{3}} = x$ by $10$ .

$\frac{10 y^{\frac{1}{3}}}{10} = \frac{x}{10}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor.

$\frac{10 y^{\frac{1}{3}}}{10} = \frac{x}{10}$

Step 2.2.2.2

Divide $y^{\frac{1}{3}}$ by $1$ .

$y^{\frac{1}{3}} = \frac{x}{10}$

Step 2.3

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

${(y^{\frac{1}{3}})}^{3} = {(\frac{x}{10})}^{3}$

Step 2.4

Simplify the exponent.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

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

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Step 2.4.1.1.1

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

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Step 2.4.1.1.1.1

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

$y^{\frac{1}{3} \cdot 3} = {(\frac{x}{10})}^{3}$

Step 2.4.1.1.1.2

Cancel the common factor of $3$ .

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Step 2.4.1.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(\frac{x}{10})}^{3}$

Step 2.4.1.1.1.2.2

Rewrite the expression.

$y^{1} = {(\frac{x}{10})}^{3}$

Step 2.4.1.1.2

Simplify.

$y = {(\frac{x}{10})}^{3}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify ${(\frac{x}{10})}^{3}$ .

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Step 2.4.2.1.1

Apply the product rule to $\frac{x}{10}$ .

$y = \frac{x^{3}}{10^{3}}$

Step 2.4.2.1.2

Raise $10$ to the power of $3$ .

$y = \frac{x^{3}}{1000}$

Step 3

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

$f^{- 1} (x) = \frac{x^{3}}{1000}$

Step 4

Verify if $f^{- 1} (x) = \frac{x^{3}}{1000}$ is the inverse of $f (x) = 10 x^{\frac{1}{3}}$ .

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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 $f^{- 1} (10 x^{\frac{1}{3}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (10 x^{\frac{1}{3}}) = \frac{{(10 x^{\frac{1}{3}})}^{3}}{1000}$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Apply the product rule to $10 x^{\frac{1}{3}}$ .

$f^{- 1} (10 x^{\frac{1}{3}}) = \frac{10^{3} {(x^{\frac{1}{3}})}^{3}}{1000}$

Step 4.2.3.2

Raise $10$ to the power of $3$ .

$f^{- 1} (10 x^{\frac{1}{3}}) = \frac{1000 {(x^{\frac{1}{3}})}^{3}}{1000}$

Step 4.2.3.3

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

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Step 4.2.3.3.1

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

$f^{- 1} (10 x^{\frac{1}{3}}) = \frac{1000 x^{\frac{1}{3} \cdot 3}}{1000}$

Step 4.2.3.3.2

Cancel the common factor of $3$ .

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Step 4.2.3.3.2.1

Cancel the common factor.

$f^{- 1} (10 x^{\frac{1}{3}}) = \frac{1000 x^{\frac{1}{3} \cdot 3}}{1000}$

Step 4.2.3.3.2.2

Rewrite the expression.

$f^{- 1} (10 x^{\frac{1}{3}}) = \frac{1000 x}{1000}$

Step 4.2.3.4

Simplify.

$f^{- 1} (10 x^{\frac{1}{3}}) = \frac{1000 x}{1000}$

Step 4.2.4

Cancel the common factor of $1000$ .

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Step 4.2.4.1

Cancel the common factor.

$f^{- 1} (10 x^{\frac{1}{3}}) = \frac{1000 x}{1000}$

Step 4.2.4.2

Divide $x$ by $1$ .

$f^{- 1} (10 x^{\frac{1}{3}}) = x$

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 $f (\frac{x^{3}}{1000})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{3}}{1000}) = 10 {(\frac{x^{3}}{1000})}^{\frac{1}{3}}$

Step 4.3.3

Apply the product rule to $\frac{x^{3}}{1000}$ .

$f (\frac{x^{3}}{1000}) = 10 (\frac{{(x^{3})}^{\frac{1}{3}}}{1000^{\frac{1}{3}}})$

Step 4.3.4

Simplify the numerator.

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Step 4.3.4.1

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

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Step 4.3.4.1.1

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

$f (\frac{x^{3}}{1000}) = 10 (\frac{x^{3 (\frac{1}{3})}}{1000^{\frac{1}{3}}})$

Step 4.3.4.1.2

Cancel the common factor of $3$ .

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Step 4.3.4.1.2.1

Cancel the common factor.

$f (\frac{x^{3}}{1000}) = 10 (\frac{x^{3 (\frac{1}{3})}}{1000^{\frac{1}{3}}})$

Step 4.3.4.1.2.2

Rewrite the expression.

$f (\frac{x^{3}}{1000}) = 10 (\frac{x}{1000^{\frac{1}{3}}})$

Step 4.3.4.2

Simplify.

$f (\frac{x^{3}}{1000}) = 10 (\frac{x}{1000^{\frac{1}{3}}})$

Step 4.3.5

Simplify the denominator.

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Step 4.3.5.1

Rewrite $1000$ as $10^{3}$ .

$f (\frac{x^{3}}{1000}) = 10 (\frac{x}{{(10^{3})}^{\frac{1}{3}}})$

Step 4.3.5.2

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

$f (\frac{x^{3}}{1000}) = 10 (\frac{x}{10^{3 (\frac{1}{3})}})$

Step 4.3.5.3

Cancel the common factor of $3$ .

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Step 4.3.5.3.1

Cancel the common factor.

$f (\frac{x^{3}}{1000}) = 10 (\frac{x}{10^{3 (\frac{1}{3})}})$

Step 4.3.5.3.2

Rewrite the expression.

$f (\frac{x^{3}}{1000}) = 10 (\frac{x}{10})$

Step 4.3.5.4

Evaluate the exponent.

$f (\frac{x^{3}}{1000}) = 10 (\frac{x}{10})$

Step 4.3.6

Cancel the common factor of $10$ .

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Step 4.3.6.1

Cancel the common factor.

$f (\frac{x^{3}}{1000}) = 10 (\frac{x}{10})$

Step 4.3.6.2

Rewrite the expression.

$f (\frac{x^{3}}{1000}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{3}}{1000}$ is the inverse of $f (x) = 10 x^{\frac{1}{3}}$ .

$f^{- 1} (x) = \frac{x^{3}}{1000}$