Algebra Examples

$\sqrt{\frac{16 u^{7} v^{2}}{4 u v^{5}}}$

Step 1

Reduce the expression $\frac{16 u^{7} v^{2}}{4 u v^{5}}$ by cancelling the common factors.

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Step 1.1

Factor $4$ out of $16 u^{7} v^{2}$ .

$\sqrt{\frac{4 (4 u^{7} v^{2})}{4 u v^{5}}}$

Step 1.2

Factor $4$ out of $4 u v^{5}$ .

$\sqrt{\frac{4 (4 u^{7} v^{2})}{4 (u v^{5})}}$

Step 1.3

Cancel the common factor.

$\sqrt{\frac{4 (4 u^{7} v^{2})}{4 (u v^{5})}}$

Step 1.4

Rewrite the expression.

$\sqrt{\frac{4 u^{7} v^{2}}{u v^{5}}}$

Step 2

Cancel the common factor of $u^{7}$ and $u$ .

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Step 2.1

Factor $u$ out of $4 u^{7} v^{2}$ .

$\sqrt{\frac{u (4 u^{6} v^{2})}{u v^{5}}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $u$ out of $u v^{5}$ .

$\sqrt{\frac{u (4 u^{6} v^{2})}{u (v^{5})}}$

Step 2.2.2

Cancel the common factor.

$\sqrt{\frac{u (4 u^{6} v^{2})}{u v^{5}}}$

Step 2.2.3

Rewrite the expression.

$\sqrt{\frac{4 u^{6} v^{2}}{v^{5}}}$

Step 3

Cancel the common factor of $v^{2}$ and $v^{5}$ .

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Step 3.1

Factor $v^{2}$ out of $4 u^{6} v^{2}$ .

$\sqrt{\frac{v^{2} (4 u^{6})}{v^{5}}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $v^{2}$ out of $v^{5}$ .

$\sqrt{\frac{v^{2} (4 u^{6})}{v^{2} v^{3}}}$

Step 3.2.2

Cancel the common factor.

$\sqrt{\frac{v^{2} (4 u^{6})}{v^{2} v^{3}}}$

Step 3.2.3

Rewrite the expression.

$\sqrt{\frac{4 u^{6}}{v^{3}}}$

Step 4

Rewrite $\frac{4 u^{6}}{v^{3}}$ as ${(\frac{2 u^{3}}{v})}^{2} \frac{1}{v}$ .

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Step 4.1

Factor the perfect power ${(2 u^{3})}^{2}$ out of $4 u^{6}$ .

$\sqrt{\frac{{(2 u^{3})}^{2} \cdot 1}{v^{3}}}$

Step 4.2

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

$\sqrt{\frac{{(2 u^{3})}^{2} \cdot 1}{v^{2} v}}$

Step 4.3

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

$\sqrt{{(\frac{2 u^{3}}{v})}^{2} \frac{1}{v}}$

Step 5

Pull terms out from under the radical.

$\frac{2 u^{3}}{v} \sqrt{\frac{1}{v}}$

Step 6

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

$\frac{2 u^{3}}{v} \cdot \frac{\sqrt{1}}{\sqrt{v}}$

Step 7

Combine.

$\frac{2 u^{3} \sqrt{1}}{v \sqrt{v}}$

Step 8

Simplify the numerator.

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Step 8.1

Any root of $1$ is $1$ .

$\frac{2 u^{3} \cdot 1}{v \sqrt{v}}$

Step 8.2

Multiply $2$ by $1$ .

$\frac{2 u^{3}}{v \sqrt{v}}$

Step 9

Multiply $\frac{2 u^{3}}{v \sqrt{v}}$ by $\frac{\sqrt{v}}{\sqrt{v}}$ .

$\frac{2 u^{3}}{v \sqrt{v}} \cdot \frac{\sqrt{v}}{\sqrt{v}}$

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{2 u^{3}}{v \sqrt{v}}$ by $\frac{\sqrt{v}}{\sqrt{v}}$ .

$\frac{2 u^{3} \sqrt{v}}{v \sqrt{v} \sqrt{v}}$

Step 10.2

Move $\sqrt{v}$ .

$\frac{2 u^{3} \sqrt{v}}{v (\sqrt{v} \sqrt{v})}$

Step 10.3

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

$\frac{2 u^{3} \sqrt{v}}{v ({\sqrt{v}}^{1} \sqrt{v})}$

Step 10.4

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

$\frac{2 u^{3} \sqrt{v}}{v ({\sqrt{v}}^{1} {\sqrt{v}}^{1})}$

Step 10.5

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

$\frac{2 u^{3} \sqrt{v}}{v {\sqrt{v}}^{1 + 1}}$

Step 10.6

Add $1$ and $1$ .

$\frac{2 u^{3} \sqrt{v}}{v {\sqrt{v}}^{2}}$

Step 10.7

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

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Step 10.7.1

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

$\frac{2 u^{3} \sqrt{v}}{v {(v^{\frac{1}{2}})}^{2}}$

Step 10.7.2

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

$\frac{2 u^{3} \sqrt{v}}{v \cdot v^{\frac{1}{2} \cdot 2}}$

Step 10.7.3

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

$\frac{2 u^{3} \sqrt{v}}{v \cdot v^{\frac{2}{2}}}$

Step 10.7.4

Cancel the common factor of $2$ .

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Step 10.7.4.1

Cancel the common factor.

$\frac{2 u^{3} \sqrt{v}}{v \cdot v^{\frac{2}{2}}}$

Step 10.7.4.2

Rewrite the expression.

$\frac{2 u^{3} \sqrt{v}}{v \cdot v^{1}}$

Step 10.7.5

Simplify.

$\frac{2 u^{3} \sqrt{v}}{v \cdot v}$

Step 11

Multiply $v$ by $v$ .

$\frac{2 u^{3} \sqrt{v}}{v^{2}}$