Algebra Examples

$\frac{\sqrt{3 x^{4}}}{\sqrt{2 x^{2} y^{5}}}$

Step 1

Combine $\sqrt{3 x^{4}}$ and $\sqrt{2 x^{2} y^{5}}$ into a single radical.

$\sqrt{\frac{3 x^{4}}{2 x^{2} y^{5}}}$

Step 2

Reduce the expression $\frac{3 x^{4}}{2 x^{2} y^{5}}$ by cancelling the common factors.

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Step 2.1

Factor $x^{2}$ out of $3 x^{4}$ .

$\sqrt{\frac{x^{2} (3 x^{2})}{2 x^{2} y^{5}}}$

Step 2.2

Factor $x^{2}$ out of $2 x^{2} y^{5}$ .

$\sqrt{\frac{x^{2} (3 x^{2})}{x^{2} (2 y^{5})}}$

Step 2.3

Cancel the common factor.

$\sqrt{\frac{x^{2} (3 x^{2})}{x^{2} (2 y^{5})}}$

Step 2.4

Rewrite the expression.

$\sqrt{\frac{3 x^{2}}{2 y^{5}}}$

Step 3

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

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Step 3.1

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

$\sqrt{\frac{x^{2} \cdot 3}{2 y^{5}}}$

Step 3.2

Factor the perfect power ${(y^{2})}^{2}$ out of $2 y^{5}$ .

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

Step 3.3

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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

Step 6

Combine.

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

Step 7

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

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

Step 8

Combine and simplify the denominator.

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Step 8.1

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

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

Step 8.2

Move $\sqrt{2 y}$ .

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

Step 8.3

Raise $\sqrt{2 y}$ to the power of $1$ .

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

Step 8.4

Raise $\sqrt{2 y}$ to the power of $1$ .

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

Step 8.5

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

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

Step 8.6

Add $1$ and $1$ .

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

Step 8.7

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

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Step 8.7.1

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

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

Step 8.7.2

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

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

Step 8.7.3

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

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

Step 8.7.4

Cancel the common factor of $2$ .

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Step 8.7.4.1

Cancel the common factor.

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

Step 8.7.4.2

Rewrite the expression.

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

Step 8.7.5

Simplify.

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

Step 9

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

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Step 9.1

Move $y$ .

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

Step 9.2

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

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Step 9.2.1

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

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

Step 9.2.2

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

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

Step 9.3

Add $1$ and $2$ .

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

Step 10

Simplify the numerator.

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Step 10.1

Combine using the product rule for radicals.

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

Step 10.2

Multiply $3$ by $2$ .

$\frac{x \sqrt{6 y}}{y^{3} \cdot 2}$

Step 11

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

$\frac{x \sqrt{6 y}}{2 y^{3}}$