Algebra Examples

$4 x^{\frac{1}{3}}$

Step 1

Interchange the variables.

$x = 4 y^{\frac{1}{3}}$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $4 y^{\frac{1}{3}} = x$ .

$4 y^{\frac{1}{3}} = x$

Step 2.2

Divide each term in $4 y^{\frac{1}{3}} = x$ by $4$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in $4 y^{\frac{1}{3}} = x$ by $4$ .

$\frac{4 y^{\frac{1}{3}}}{4} = \frac{x}{4}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Cancel the common factor.

$\frac{4 y^{\frac{1}{3}}}{4} = \frac{x}{4}$

Step 2.2.2.2

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

$y^{\frac{1}{3}} = \frac{x}{4}$

Step 2.3

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

${(y^{\frac{1}{3}})}^{3} = {(\frac{x}{4})}^{3}$

Step 2.4

Simplify the exponent.

Tap for more steps...

Step 2.4.1

Simplify the left side.

Tap for more steps...

Step 2.4.1.1

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

Tap for more steps...

Step 2.4.1.1.1

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

Tap for more steps...

Step 2.4.1.1.1.1

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

$y^{\frac{1}{3} \cdot 3} = {(\frac{x}{4})}^{3}$

Step 2.4.1.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.4.1.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(\frac{x}{4})}^{3}$

Step 2.4.1.1.1.2.2

Rewrite the expression.

$y^{1} = {(\frac{x}{4})}^{3}$

Step 2.4.1.1.2

Simplify.

$y = {(\frac{x}{4})}^{3}$

Step 2.4.2

Simplify the right side.

Tap for more steps...

Step 2.4.2.1

Simplify ${(\frac{x}{4})}^{3}$ .

Tap for more steps...

Step 2.4.2.1.1

Apply the product rule to $\frac{x}{4}$ .

$y = \frac{x^{3}}{4^{3}}$

Step 2.4.2.1.2

Raise $4$ to the power of $3$ .

$y = \frac{x^{3}}{64}$

Step 3

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

$f^{- 1} (x) = \frac{x^{3}}{64}$

Step 4

Verify if $f^{- 1} (x) = \frac{x^{3}}{64}$ is the inverse of $f (x) = 4 x^{\frac{1}{3}}$ .

Tap for more steps...

Step 4.1

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

Step 4.2

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

Tap for more steps...

Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (4 x^{\frac{1}{3}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (4 x^{\frac{1}{3}}) = \frac{{(4 x^{\frac{1}{3}})}^{3}}{64}$

Step 4.2.3

Simplify the numerator.

Tap for more steps...

Step 4.2.3.1

Apply the product rule to $4 x^{\frac{1}{3}}$ .

$f^{- 1} (4 x^{\frac{1}{3}}) = \frac{4^{3} {(x^{\frac{1}{3}})}^{3}}{64}$

Step 4.2.3.2

Raise $4$ to the power of $3$ .

$f^{- 1} (4 x^{\frac{1}{3}}) = \frac{64 {(x^{\frac{1}{3}})}^{3}}{64}$

Step 4.2.3.3

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

Tap for more steps...

Step 4.2.3.3.1

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

$f^{- 1} (4 x^{\frac{1}{3}}) = \frac{64 x^{\frac{1}{3} \cdot 3}}{64}$

Step 4.2.3.3.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.3.3.2.1

Cancel the common factor.

$f^{- 1} (4 x^{\frac{1}{3}}) = \frac{64 x^{\frac{1}{3} \cdot 3}}{64}$

Step 4.2.3.3.2.2

Rewrite the expression.

$f^{- 1} (4 x^{\frac{1}{3}}) = \frac{64 x}{64}$

Step 4.2.3.4

Simplify.

$f^{- 1} (4 x^{\frac{1}{3}}) = \frac{64 x}{64}$

Step 4.2.4

Cancel the common factor of $64$ .

Tap for more steps...

Step 4.2.4.1

Cancel the common factor.

$f^{- 1} (4 x^{\frac{1}{3}}) = \frac{64 x}{64}$

Step 4.2.4.2

Divide $x$ by $1$ .

$f^{- 1} (4 x^{\frac{1}{3}}) = x$

Step 4.3

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

Tap for more steps...

Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\frac{x^{3}}{64})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{3}}{64}) = 4 {(\frac{x^{3}}{64})}^{\frac{1}{3}}$

Step 4.3.3

Apply the product rule to $\frac{x^{3}}{64}$ .

$f (\frac{x^{3}}{64}) = 4 (\frac{{(x^{3})}^{\frac{1}{3}}}{64^{\frac{1}{3}}})$

Step 4.3.4

Simplify the numerator.

Tap for more steps...

Step 4.3.4.1

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

Tap for more steps...

Step 4.3.4.1.1

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

$f (\frac{x^{3}}{64}) = 4 (\frac{x^{3 (\frac{1}{3})}}{64^{\frac{1}{3}}})$

Step 4.3.4.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.3.4.1.2.1

Cancel the common factor.

$f (\frac{x^{3}}{64}) = 4 (\frac{x^{3 (\frac{1}{3})}}{64^{\frac{1}{3}}})$

Step 4.3.4.1.2.2

Rewrite the expression.

$f (\frac{x^{3}}{64}) = 4 (\frac{x}{64^{\frac{1}{3}}})$

Step 4.3.4.2

Simplify.

$f (\frac{x^{3}}{64}) = 4 (\frac{x}{64^{\frac{1}{3}}})$

Step 4.3.5

Simplify the denominator.

Tap for more steps...

Step 4.3.5.1

Rewrite $64$ as $4^{3}$ .

$f (\frac{x^{3}}{64}) = 4 (\frac{x}{{(4^{3})}^{\frac{1}{3}}})$

Step 4.3.5.2

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

$f (\frac{x^{3}}{64}) = 4 (\frac{x}{4^{3 (\frac{1}{3})}})$

Step 4.3.5.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.3.5.3.1

Cancel the common factor.

$f (\frac{x^{3}}{64}) = 4 (\frac{x}{4^{3 (\frac{1}{3})}})$

Step 4.3.5.3.2

Rewrite the expression.

$f (\frac{x^{3}}{64}) = 4 (\frac{x}{4})$

Step 4.3.5.4

Evaluate the exponent.

$f (\frac{x^{3}}{64}) = 4 (\frac{x}{4})$

Step 4.3.6

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.3.6.1

Cancel the common factor.

$f (\frac{x^{3}}{64}) = 4 (\frac{x}{4})$

Step 4.3.6.2

Rewrite the expression.

$f (\frac{x^{3}}{64}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{3}}{64}$ is the inverse of $f (x) = 4 x^{\frac{1}{3}}$ .

$f^{- 1} (x) = \frac{x^{3}}{64}$