Simplify cube root of (x^5)/(y^7)

$\sqrt[3]{\frac{x^{5}}{y^{7}}}$

Step 1

Rewrite $\frac{x^{5}}{y^{7}}$ as ${(\frac{x}{y^{2}})}^{3} \frac{x^{2}}{y}$ .

Factor the perfect power $x^{3}$ out of $x^{5}$ .

$\sqrt[3]{\frac{x^{3} x^{2}}{y^{7}}}$

Factor the perfect power ${(y^{2})}^{3}$ out of $y^{7}$ .

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

Rearrange the fraction $\frac{x^{3} x^{2}}{{(y^{2})}^{3} y}$ .

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

Step 2

Pull terms out from under the radical.

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

Step 3

Rewrite $\sqrt[3]{\frac{x^{2}}{y}}$ as $\frac{\sqrt[3]{x^{2}}}{\sqrt[3]{y}}$ .

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

Step 4

Combine.

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

Step 5

Multiply $\frac{x \sqrt[3]{x^{2}}}{y^{2} \sqrt[3]{y}}$ by $\frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$ .

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

Step 6

Combine and simplify the denominator.

Multiply $\frac{x \sqrt[3]{x^{2}}}{y^{2} \sqrt[3]{y}}$ by $\frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$ .

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

Move $\sqrt[3]{y}$ .

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

Raise $\sqrt[3]{y}$ to the power of $1$ .

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

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

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

Add $1$ and $2$ .

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

Rewrite ${\sqrt[3]{y}}^{3}$ as $y$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y}$ as $y^{\frac{1}{3}}$ .

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 7

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

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

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

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

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

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

Add $2$ and $1$ .

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

Step 8

Simplify the numerator.

Rewrite ${\sqrt[3]{y}}^{2}$ as $\sqrt[3]{y^{2}}$ .

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

Combine using the product rule for radicals.

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

Simplify cube root of (x^5)/(y^7)

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