Algebra Examples

$\sqrt{\frac{5 m}{7 n^{2}}}$

Step 1

Rewrite $\frac{5 m}{7 n^{2}}$ as ${(\frac{1}{n})}^{2} \frac{5 m}{7}$ .

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Step 1.1

Factor the perfect power $1^{2}$ out of $5 m$ .

$\sqrt{\frac{1^{2} (5 m)}{7 n^{2}}}$

Step 1.2

Factor the perfect power $n^{2}$ out of $7 n^{2}$ .

$\sqrt{\frac{1^{2} (5 m)}{n^{2} \cdot 7}}$

Step 1.3

Rearrange the fraction $\frac{1^{2} (5 m)}{n^{2} \cdot 7}$ .

$\sqrt{{(\frac{1}{n})}^{2} \frac{5 m}{7}}$

Step 2

Pull terms out from under the radical.

$\frac{1}{n} \sqrt{\frac{5 m}{7}}$

Step 3

Rewrite $\sqrt{\frac{5 m}{7}}$ as $\frac{\sqrt{5 m}}{\sqrt{7}}$ .

$\frac{1}{n} \cdot \frac{\sqrt{5 m}}{\sqrt{7}}$

Step 4

Multiply $\frac{\sqrt{5 m}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{1}{n} (\frac{\sqrt{5 m}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{\sqrt{5 m}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{1}{n} \cdot \frac{\sqrt{5 m} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 5.2

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{1}{n} \cdot \frac{\sqrt{5 m} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 5.3

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{1}{n} \cdot \frac{\sqrt{5 m} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 5.4

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

$\frac{1}{n} \cdot \frac{\sqrt{5 m} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 5.5

Add $1$ and $1$ .

$\frac{1}{n} \cdot \frac{\sqrt{5 m} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 5.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{1}{n} \cdot \frac{\sqrt{5 m} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 5.6.2

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

$\frac{1}{n} \cdot \frac{\sqrt{5 m} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 5.6.3

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

$\frac{1}{n} \cdot \frac{\sqrt{5 m} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 5.6.4

Cancel the common factor of $2$ .

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Step 5.6.4.1

Cancel the common factor.

$\frac{1}{n} \cdot \frac{\sqrt{5 m} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 5.6.4.2

Rewrite the expression.

$\frac{1}{n} \cdot \frac{\sqrt{5 m} \sqrt{7}}{7^{1}}$

Step 5.6.5

Evaluate the exponent.

$\frac{1}{n} \cdot \frac{\sqrt{5 m} \sqrt{7}}{7}$

Step 6

Simplify the numerator.

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Step 6.1

Combine using the product rule for radicals.

$\frac{1}{n} \cdot \frac{\sqrt{5 m \cdot 7}}{7}$

Step 6.2

Multiply $7$ by $5$ .

$\frac{1}{n} \cdot \frac{\sqrt{35 m}}{7}$

Step 7

Combine fractions.

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Step 7.1

Multiply $\frac{1}{n}$ by $\frac{\sqrt{35 m}}{7}$ .

$\frac{\sqrt{35 m}}{n \cdot 7}$

Step 7.2

Move $7$ to the left of $n$ .

$\frac{\sqrt{35 m}}{7 n}$