Algebra Examples

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

Step 1

Combine $\sqrt{5 m}$ and $\sqrt{5 n}$ into a single radical.

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

Step 2

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

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Step 2.1

Cancel the common factor.

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

Step 2.2

Rewrite the expression.

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

Step 3

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

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

Step 4

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

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

Step 5

Combine and simplify the denominator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Add $1$ and $1$ .

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

Step 5.6

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

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Step 5.6.1

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

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

Step 5.6.2

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

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

Step 5.6.3

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

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

Step 5.6.4

Cancel the common factor of $2$ .

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Step 5.6.4.1

Cancel the common factor.

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

Step 5.6.4.2

Rewrite the expression.

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

Step 5.6.5

Simplify.

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

Step 6

Combine using the product rule for radicals.

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