Algebra Examples

$\frac{\sqrt{x^{5}}}{\sqrt{3 x y^{3}}}$

Step 1

Combine $\sqrt{x^{5}}$ and $\sqrt{3 x y^{3}}$ into a single radical.

$\sqrt{\frac{x^{5}}{3 x y^{3}}}$

Step 2

Reduce the expression $\frac{x^{5}}{3 x y^{3}}$ by cancelling the common factors.

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Step 2.1

Factor $x$ out of $x^{5}$ .

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

Step 2.2

Factor $x$ out of $3 x y^{3}$ .

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

Step 2.3

Cancel the common factor.

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

Step 2.4

Rewrite the expression.

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

Step 3

Rewrite $\frac{x^{4}}{3 y^{3}}$ as ${(\frac{x^{2}}{y})}^{2} \frac{1}{3 y}$ .

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Step 3.1

Factor the perfect power ${(x^{2})}^{2}$ out of $x^{4}$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

Rewrite $\sqrt{\frac{1}{3 y}}$ as $\frac{\sqrt{1}}{\sqrt{3 y}}$ .

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

Step 6

Combine.

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

Step 7

Any root of $1$ is $1$ .

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

Step 8

Multiply $x^{2}$ by $1$ .

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

Step 9

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

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

Step 10

Combine and simplify the denominator.

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Step 10.1

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

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

Step 10.2

Move $\sqrt{3 y}$ .

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

Step 10.3

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

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

Step 10.4

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

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

Step 10.5

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

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

Step 10.6

Add $1$ and $1$ .

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

Step 10.7

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

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Step 10.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 y}$ as ${(3 y)}^{\frac{1}{2}}$ .

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

Step 10.7.2

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

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

Step 10.7.3

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

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

Step 10.7.4

Cancel the common factor of $2$ .

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Step 10.7.4.1

Cancel the common factor.

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

Step 10.7.4.2

Rewrite the expression.

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

Step 10.7.5

Simplify.

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

Step 11

Multiply $y$ by $y$ by adding the exponents.

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Step 11.1

Move $y$ .

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

Step 11.2

Multiply $y$ by $y$ .

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

Step 12

Move $3$ to the left of $y^{2}$ .

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