Algebra Examples

$\sqrt{\frac{x^{2}}{12}}$

Step 1

Rewrite $\frac{x^{2}}{12}$ as ${(\frac{x}{2})}^{2} \frac{1}{3}$ .

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Step 1.1

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

$\sqrt{\frac{x^{2} \cdot 1}{12}}$

Step 1.2

Factor the perfect power $2^{2}$ out of $12$ .

$\sqrt{\frac{x^{2} \cdot 1}{2^{2} \cdot 3}}$

Step 1.3

Rearrange the fraction $\frac{x^{2} \cdot 1}{2^{2} \cdot 3}$ .

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

Step 2

Pull terms out from under the radical.

$| \frac{x}{2} | \sqrt{\frac{1}{3}}$

Step 3

Remove non-negative terms from the absolute value.

$\frac{| x |}{2} \sqrt{\frac{1}{3}}$

Step 4

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$\frac{| x |}{2} \cdot \frac{\sqrt{1}}{\sqrt{3}}$

Step 5

Any root of $1$ is $1$ .

$\frac{| x |}{2} \cdot \frac{1}{\sqrt{3}}$

Step 6

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{| x |}{2} (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{| x |}{2} \cdot \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 7.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{| x |}{2} \cdot \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 7.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{| x |}{2} \cdot \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 7.4

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

$\frac{| x |}{2} \cdot \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

$\frac{| x |}{2} \cdot \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 7.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 7.6.2

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

$\frac{| x |}{2} \cdot \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 7.6.3

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

$\frac{| x |}{2} \cdot \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$\frac{| x |}{2} \cdot \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 7.6.4.2

Rewrite the expression.

$\frac{| x |}{2} \cdot \frac{\sqrt{3}}{3^{1}}$

Step 7.6.5

Evaluate the exponent.

$\frac{| x |}{2} \cdot \frac{\sqrt{3}}{3}$

Step 8

Multiply $\frac{| x |}{2} \cdot \frac{\sqrt{3}}{3}$ .

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Step 8.1

Multiply $\frac{| x |}{2}$ by $\frac{\sqrt{3}}{3}$ .

$\frac{| x | \sqrt{3}}{2 \cdot 3}$

Step 8.2

Multiply $2$ by $3$ .

$\frac{| x | \sqrt{3}}{6}$