Precalculus Examples

$\frac{\sqrt[3]{54 x y^{2}}}{\sqrt[3]{2 x y^{- 1}}}$

Step 1

Combine $\sqrt[3]{54 x y^{2}}$ and $\sqrt[3]{2 x y^{- 1}}$ into a single radical.

$\sqrt[3]{\frac{54 x y^{2}}{2 x y^{- 1}}}$

Step 2

Reduce the expression $\frac{54 x y^{2}}{2 x y^{- 1}}$ by cancelling the common factors.

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Step 2.1

Factor $2$ out of $54 x y^{2}$ .

$\sqrt[3]{\frac{2 (27 x y^{2})}{2 x y^{- 1}}}$

Step 2.2

Factor $2$ out of $2 x y^{- 1}$ .

$\sqrt[3]{\frac{2 (27 x y^{2})}{2 (x y^{- 1})}}$

Step 2.3

Cancel the common factor.

$\sqrt[3]{\frac{2 (27 x y^{2})}{2 (x y^{- 1})}}$

Step 2.4

Rewrite the expression.

$\sqrt[3]{\frac{27 x y^{2}}{x y^{- 1}}}$

Step 3

Move $y^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\sqrt[3]{\frac{27 x y^{2} y}{x}}$

Step 4

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 4.1

Move $y$ .

$\sqrt[3]{\frac{27 x (y \cdot y^{2})}{x}}$

Step 4.2

Multiply $y$ by $y^{2}$ .

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Step 4.2.1

Raise $y$ to the power of $1$ .

$\sqrt[3]{\frac{27 x (y^{1} y^{2})}{x}}$

Step 4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt[3]{\frac{27 x y^{1 + 2}}{x}}$

Step 4.3

Add $1$ and $2$ .

$\sqrt[3]{\frac{27 x y^{3}}{x}}$

Step 5

Cancel the common factor of $x$ .

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Step 5.1

Cancel the common factor.

$\sqrt[3]{\frac{27 x y^{3}}{x}}$

Step 5.2

Divide $27 y^{3}$ by $1$ .

$\sqrt[3]{27 y^{3}}$

Step 6

Rewrite $27 y^{3}$ as ${(3 y)}^{3}$ .

$\sqrt[3]{{(3 y)}^{3}}$

Step 7

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

$3 y$