Algebra Examples

$\sqrt{\frac{4 m^{4}}{5 m^{2} n^{4}}}$

Step 1

Reduce the expression $\frac{4 m^{4}}{5 m^{2} n^{4}}$ by cancelling the common factors.

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Step 1.1

Factor $m^{2}$ out of $4 m^{4}$ .

$\sqrt{\frac{m^{2} (4 m^{2})}{5 m^{2} n^{4}}}$

Step 1.2

Factor $m^{2}$ out of $5 m^{2} n^{4}$ .

$\sqrt{\frac{m^{2} (4 m^{2})}{m^{2} (5 n^{4})}}$

Step 1.3

Cancel the common factor.

$\sqrt{\frac{m^{2} (4 m^{2})}{m^{2} (5 n^{4})}}$

Step 1.4

Rewrite the expression.

$\sqrt{\frac{4 m^{2}}{5 n^{4}}}$

Step 2

Rewrite $\frac{4 m^{2}}{5 n^{4}}$ as ${(\frac{2 m}{n^{2}})}^{2} \frac{1}{5}$ .

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Step 2.1

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

$\sqrt{\frac{{(2 m)}^{2} \cdot 1}{5 n^{4}}}$

Step 2.2

Factor the perfect power ${(n^{2})}^{2}$ out of $5 n^{4}$ .

$\sqrt{\frac{{(2 m)}^{2} \cdot 1}{{(n^{2})}^{2} \cdot 5}}$

Step 2.3

Rearrange the fraction $\frac{{(2 m)}^{2} \cdot 1}{{(n^{2})}^{2} \cdot 5}$ .

$\sqrt{{(\frac{2 m}{n^{2}})}^{2} \frac{1}{5}}$

Step 3

Pull terms out from under the radical.

$\frac{2 m}{n^{2}} \sqrt{\frac{1}{5}}$

Step 4

Rewrite $\sqrt{\frac{1}{5}}$ as $\frac{\sqrt{1}}{\sqrt{5}}$ .

$\frac{2 m}{n^{2}} \cdot \frac{\sqrt{1}}{\sqrt{5}}$

Step 5

Combine.

$\frac{2 m \sqrt{1}}{n^{2} \sqrt{5}}$

Step 6

Simplify the numerator.

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Step 6.1

Any root of $1$ is $1$ .

$\frac{2 m \cdot 1}{n^{2} \sqrt{5}}$

Step 6.2

Multiply $2$ by $1$ .

$\frac{2 m}{n^{2} \sqrt{5}}$

Step 7

Multiply $\frac{2 m}{n^{2} \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{2 m}{n^{2} \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{2 m}{n^{2} \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{2 m \sqrt{5}}{n^{2} \sqrt{5} \sqrt{5}}$

Step 8.2

Move $\sqrt{5}$ .

$\frac{2 m \sqrt{5}}{n^{2} (\sqrt{5} \sqrt{5})}$

Step 8.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{2 m \sqrt{5}}{n^{2} ({\sqrt{5}}^{1} \sqrt{5})}$

Step 8.4

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{2 m \sqrt{5}}{n^{2} ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}$

Step 8.5

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

$\frac{2 m \sqrt{5}}{n^{2} {\sqrt{5}}^{1 + 1}}$

Step 8.6

Add $1$ and $1$ .

$\frac{2 m \sqrt{5}}{n^{2} {\sqrt{5}}^{2}}$

Step 8.7

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 8.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{2 m \sqrt{5}}{n^{2} {(5^{\frac{1}{2}})}^{2}}$

Step 8.7.2

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

$\frac{2 m \sqrt{5}}{n^{2} \cdot 5^{\frac{1}{2} \cdot 2}}$

Step 8.7.3

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

$\frac{2 m \sqrt{5}}{n^{2} \cdot 5^{\frac{2}{2}}}$

Step 8.7.4

Cancel the common factor of $2$ .

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Step 8.7.4.1

Cancel the common factor.

$\frac{2 m \sqrt{5}}{n^{2} \cdot 5^{\frac{2}{2}}}$

Step 8.7.4.2

Rewrite the expression.

$\frac{2 m \sqrt{5}}{n^{2} \cdot 5^{1}}$

Step 8.7.5

Evaluate the exponent.

$\frac{2 m \sqrt{5}}{n^{2} \cdot 5}$

Step 9

Move $5$ to the left of $n^{2}$ .

$\frac{2 m \sqrt{5}}{5 n^{2}}$