Algebra Examples

$\frac{2}{\sqrt{18 q^{3}}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $18 q^{3}$ as ${(3 q)}^{2} \cdot (2 q)$ .

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Step 1.1.1

Factor $9$ out of $18$ .

$\frac{2}{\sqrt{9 (2) q^{3}}}$

Step 1.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{2}{\sqrt{3^{2} \cdot 2 q^{3}}}$

Step 1.1.3

Factor out $q^{2}$ .

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

Step 1.1.4

Move $2$ .

$\frac{2}{\sqrt{3^{2} q^{2} \cdot 2 q}}$

Step 1.1.5

Rewrite $3^{2} q^{2}$ as ${(3 q)}^{2}$ .

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

Step 1.1.6

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

$\frac{2}{3 q \sqrt{2 q}}$

Step 2

Multiply $\frac{2}{3 q \sqrt{2 q}}$ by $\frac{\sqrt{2 q}}{\sqrt{2 q}}$ .

$\frac{2}{3 q \sqrt{2 q}} \cdot \frac{\sqrt{2 q}}{\sqrt{2 q}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{2}{3 q \sqrt{2 q}}$ by $\frac{\sqrt{2 q}}{\sqrt{2 q}}$ .

$\frac{2 \sqrt{2 q}}{3 q \sqrt{2 q} \sqrt{2 q}}$

Step 3.2

Move $\sqrt{2 q}$ .

$\frac{2 \sqrt{2 q}}{3 q (\sqrt{2 q} \sqrt{2 q})}$

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

$\frac{2 \sqrt{2 q}}{3 q {\sqrt{2 q}}^{1 + 1}}$

Step 3.6

Add $1$ and $1$ .

$\frac{2 \sqrt{2 q}}{3 q {\sqrt{2 q}}^{2}}$

Step 3.7

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

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Step 3.7.1

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

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

Step 3.7.2

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

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

Step 3.7.3

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

$\frac{2 \sqrt{2 q}}{3 q {(2 q)}^{\frac{2}{2}}}$

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

$\frac{2 \sqrt{2 q}}{3 q {(2 q)}^{\frac{2}{2}}}$

Step 3.7.4.2

Rewrite the expression.

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

Step 3.7.5

Simplify.

$\frac{2 \sqrt{2 q}}{3 q (2 q)}$

Step 4

Multiply $q$ by $q$ by adding the exponents.

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Step 4.1

Move $q$ .

$\frac{2 \sqrt{2 q}}{3 (q \cdot q) \cdot 2}$

Step 4.2

Multiply $q$ by $q$ .

$\frac{2 \sqrt{2 q}}{3 q^{2} \cdot 2}$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Cancel the common factor.

$\frac{2 \sqrt{2 q}}{3 q^{2} \cdot 2}$

Step 5.2

Rewrite the expression.

$\frac{\sqrt{2 q}}{3 q^{2}}$