Simplify square root of (x^6)/(y^7)

$\sqrt{\frac{x^{6}}{y^{7}}}$

Step 1

Rewrite $\frac{x^{6}}{y^{7}}$ as ${(\frac{x^{3}}{y^{3}})}^{2} \frac{1}{y}$ .

Factor the perfect power ${(x^{3})}^{2}$ out of $x^{6}$ .

$\sqrt{\frac{{(x^{3})}^{2} \cdot 1}{y^{7}}}$

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

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

Rearrange the fraction $\frac{{(x^{3})}^{2} \cdot 1}{{(y^{3})}^{2} y}$ .

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

Step 2

Pull terms out from under the radical.

$\frac{x^{3}}{y^{3}} \sqrt{\frac{1}{y}}$

Step 3

Rewrite $\sqrt{\frac{1}{y}}$ as $\frac{\sqrt{1}}{\sqrt{y}}$ .

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

Step 4

Combine.

$\frac{x^{3} \sqrt{1}}{y^{3} \sqrt{y}}$

Step 5

Any root of $1$ is $1$ .

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

Step 6

Multiply $x^{3}$ by $1$ .

$\frac{x^{3}}{y^{3} \sqrt{y}}$

Step 7

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

$\frac{x^{3}}{y^{3} \sqrt{y}} \cdot \frac{\sqrt{y}}{\sqrt{y}}$

Step 8

Combine and simplify the denominator.

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

$\frac{x^{3} \sqrt{y}}{y^{3} \sqrt{y} \sqrt{y}}$

Move $\sqrt{y}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{y}}^{2}$ as $y$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

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

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

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x^{3} \sqrt{y}}{y^{3} y^{1}}$

Simplify.

$\frac{x^{3} \sqrt{y}}{y^{3} y}$

Step 9

Multiply $y^{3}$ by $y$ by adding the exponents.

Multiply $y^{3}$ by $y$ .

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

$\frac{x^{3} \sqrt{y}}{y^{3} y^{1}}$

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

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

Add $3$ and $1$ .

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

Simplify square root of (x^6)/(y^7)

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