Algebra Examples

$\frac{a^{3}}{\sqrt{15 a^{5}}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $15 a^{5}$ as ${(a^{2})}^{2} \cdot (15 a)$ .

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Step 1.1.1

Factor out $a^{4}$ .

$\frac{a^{3}}{\sqrt{15 (a^{4} a)}}$

Step 1.1.2

Rewrite $a^{4}$ as ${(a^{2})}^{2}$ .

$\frac{a^{3}}{\sqrt{15 ({(a^{2})}^{2} a)}}$

Step 1.1.3

Reorder $15$ and ${(a^{2})}^{2}$ .

$\frac{a^{3}}{\sqrt{{(a^{2})}^{2} \cdot 15 a}}$

Step 1.1.4

Add parentheses.

$\frac{a^{3}}{\sqrt{{(a^{2})}^{2} \cdot (15 a)}}$

Step 1.2

Pull terms out from under the radical.

$\frac{a^{3}}{a^{2} \sqrt{15 a}}$

Step 2

Cancel the common factor of $a^{3}$ and $a^{2}$ .

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Step 2.1

Factor $a^{2}$ out of $a^{3}$ .

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

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $a^{2}$ out of $a^{2} \sqrt{15 a}$ .

$\frac{a^{2} a}{a^{2} (\sqrt{15 a})}$

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

$\frac{a}{\sqrt{15 a}}$

Step 3

Multiply $\frac{a}{\sqrt{15 a}}$ by $\frac{\sqrt{15 a}}{\sqrt{15 a}}$ .

$\frac{a}{\sqrt{15 a}} \cdot \frac{\sqrt{15 a}}{\sqrt{15 a}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{a}{\sqrt{15 a}}$ by $\frac{\sqrt{15 a}}{\sqrt{15 a}}$ .

$\frac{a \sqrt{15 a}}{\sqrt{15 a} \sqrt{15 a}}$

Step 4.2

Raise $\sqrt{15 a}$ to the power of $1$ .

$\frac{a \sqrt{15 a}}{{\sqrt{15 a}}^{1} \sqrt{15 a}}$

Step 4.3

Raise $\sqrt{15 a}$ to the power of $1$ .

$\frac{a \sqrt{15 a}}{{\sqrt{15 a}}^{1} {\sqrt{15 a}}^{1}}$

Step 4.4

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

$\frac{a \sqrt{15 a}}{{\sqrt{15 a}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

Rewrite ${\sqrt{15 a}}^{2}$ as $15 a$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15 a}$ as ${(15 a)}^{\frac{1}{2}}$ .

$\frac{a \sqrt{15 a}}{{({(15 a)}^{\frac{1}{2}})}^{2}}$

Step 4.6.2

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

$\frac{a \sqrt{15 a}}{{(15 a)}^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

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

$\frac{a \sqrt{15 a}}{{(15 a)}^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{a \sqrt{15 a}}{{(15 a)}^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$\frac{a \sqrt{15 a}}{{(15 a)}^{1}}$

Step 4.6.5

Simplify.

$\frac{a \sqrt{15 a}}{15 a}$

Step 5

Cancel the common factor of $a$ .

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Step 5.1

Cancel the common factor.

$\frac{a \sqrt{15 a}}{15 a}$

Step 5.2

Rewrite the expression.

$\frac{\sqrt{15 a}}{15}$