Basic Math Examples

$\sqrt{\frac{a^{5}}{45}}$

Step 1

Rewrite $\frac{a^{5}}{45}$ as ${(\frac{a^{2}}{3})}^{2} \frac{a}{5}$ .

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Step 1.1

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

$\sqrt{\frac{{(a^{2})}^{2} a}{45}}$

Step 1.2

Factor the perfect power $3^{2}$ out of $45$ .

$\sqrt{\frac{{(a^{2})}^{2} a}{3^{2} \cdot 5}}$

Step 1.3

Rearrange the fraction $\frac{{(a^{2})}^{2} a}{3^{2} \cdot 5}$ .

$\sqrt{{(\frac{a^{2}}{3})}^{2} \frac{a}{5}}$

Step 2

Pull terms out from under the radical.

$\frac{a^{2}}{3} \sqrt{\frac{a}{5}}$

Step 3

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

$\frac{a^{2}}{3} \cdot \frac{\sqrt{a}}{\sqrt{5}}$

Step 4

Combine.

$\frac{a^{2} \sqrt{a}}{3 \sqrt{5}}$

Step 5

Multiply $\frac{a^{2} \sqrt{a}}{3 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{a^{2} \sqrt{a}}{3 \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{a^{2} \sqrt{a}}{3 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{a^{2} \sqrt{a} \sqrt{5}}{3 \sqrt{5} \sqrt{5}}$

Step 6.2

Move $\sqrt{5}$ .

$\frac{a^{2} \sqrt{a} \sqrt{5}}{3 (\sqrt{5} \sqrt{5})}$

Step 6.3

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

$\frac{a^{2} \sqrt{a} \sqrt{5}}{3 ({\sqrt{5}}^{1} \sqrt{5})}$

Step 6.4

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

$\frac{a^{2} \sqrt{a} \sqrt{5}}{3 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}$

Step 6.5

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

$\frac{a^{2} \sqrt{a} \sqrt{5}}{3 {\sqrt{5}}^{1 + 1}}$

Step 6.6

Add $1$ and $1$ .

$\frac{a^{2} \sqrt{a} \sqrt{5}}{3 {\sqrt{5}}^{2}}$

Step 6.7

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

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Step 6.7.1

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

$\frac{a^{2} \sqrt{a} \sqrt{5}}{3 {(5^{\frac{1}{2}})}^{2}}$

Step 6.7.2

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

$\frac{a^{2} \sqrt{a} \sqrt{5}}{3 \cdot 5^{\frac{1}{2} \cdot 2}}$

Step 6.7.3

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

$\frac{a^{2} \sqrt{a} \sqrt{5}}{3 \cdot 5^{\frac{2}{2}}}$

Step 6.7.4

Cancel the common factor of $2$ .

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Step 6.7.4.1

Cancel the common factor.

$\frac{a^{2} \sqrt{a} \sqrt{5}}{3 \cdot 5^{\frac{2}{2}}}$

Step 6.7.4.2

Rewrite the expression.

$\frac{a^{2} \sqrt{a} \sqrt{5}}{3 \cdot 5^{1}}$

Step 6.7.5

Evaluate the exponent.

$\frac{a^{2} \sqrt{a} \sqrt{5}}{3 \cdot 5}$

Step 7

Combine using the product rule for radicals.

$\frac{a^{2} \sqrt{5 a}}{3 \cdot 5}$

Step 8

Multiply $3$ by $5$ .

$\frac{a^{2} \sqrt{5 a}}{15}$