Algebra Examples

$\sqrt{\frac{3 m}{2 n}}$

Step 1

Rewrite $\sqrt{\frac{3 m}{2 n}}$ as $\frac{\sqrt{3 m}}{\sqrt{2 n}}$ .

$\frac{\sqrt{3 m}}{\sqrt{2 n}}$

Step 2

Multiply $\frac{\sqrt{3 m}}{\sqrt{2 n}}$ by $\frac{\sqrt{2 n}}{\sqrt{2 n}}$ .

$\frac{\sqrt{3 m}}{\sqrt{2 n}} \cdot \frac{\sqrt{2 n}}{\sqrt{2 n}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt{3 m}}{\sqrt{2 n}}$ by $\frac{\sqrt{2 n}}{\sqrt{2 n}}$ .

$\frac{\sqrt{3 m} \sqrt{2 n}}{\sqrt{2 n} \sqrt{2 n}}$

Step 3.2

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

$\frac{\sqrt{3 m} \sqrt{2 n}}{{\sqrt{2 n}}^{1} \sqrt{2 n}}$

Step 3.3

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

$\frac{\sqrt{3 m} \sqrt{2 n}}{{\sqrt{2 n}}^{1} {\sqrt{2 n}}^{1}}$

Step 3.4

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

$\frac{\sqrt{3 m} \sqrt{2 n}}{{\sqrt{2 n}}^{1 + 1}}$

Step 3.5

Add $1$ and $1$ .

$\frac{\sqrt{3 m} \sqrt{2 n}}{{\sqrt{2 n}}^{2}}$

Step 3.6

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

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Step 3.6.1

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

$\frac{\sqrt{3 m} \sqrt{2 n}}{{({(2 n)}^{\frac{1}{2}})}^{2}}$

Step 3.6.2

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

$\frac{\sqrt{3 m} \sqrt{2 n}}{{(2 n)}^{\frac{1}{2} \cdot 2}}$

Step 3.6.3

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

$\frac{\sqrt{3 m} \sqrt{2 n}}{{(2 n)}^{\frac{2}{2}}}$

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

$\frac{\sqrt{3 m} \sqrt{2 n}}{{(2 n)}^{\frac{2}{2}}}$

Step 3.6.4.2

Rewrite the expression.

$\frac{\sqrt{3 m} \sqrt{2 n}}{{(2 n)}^{1}}$

Step 3.6.5

Simplify.

$\frac{\sqrt{3 m} \sqrt{2 n}}{2 n}$

Step 4

Simplify the numerator.

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Step 4.1

Combine using the product rule for radicals.

$\frac{\sqrt{3 m (2 n)}}{2 n}$

Step 4.2

Multiply $2$ by $3$ .

$\frac{\sqrt{6 m n}}{2 n}$