Algebra Examples

$\sqrt{\frac{2}{x^{2}}} \cdot \sqrt{\frac{x^{2}}{18}}$

Step 1

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

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Step 1.1

Factor the perfect power $1^{2}$ out of $2$ .

$\sqrt{\frac{1^{2} \cdot 2}{x^{2}}} \cdot \sqrt{\frac{x^{2}}{18}}$

Step 1.2

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

$\sqrt{\frac{1^{2} \cdot 2}{x^{2} \cdot 1}} \cdot \sqrt{\frac{x^{2}}{18}}$

Step 1.3

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

$\sqrt{{(\frac{1}{x})}^{2} \cdot 2} \cdot \sqrt{\frac{x^{2}}{18}}$

Step 2

Pull terms out from under the radical.

$\frac{1}{x} \sqrt{2} \cdot \sqrt{\frac{x^{2}}{18}}$

Step 3

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

$\frac{\sqrt{2}}{x} \cdot \sqrt{\frac{x^{2}}{18}}$

Step 4

Rewrite $\frac{x^{2}}{18}$ as ${(\frac{x}{3})}^{2} \frac{1}{2}$ .

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Step 4.1

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

$\frac{\sqrt{2}}{x} \cdot \sqrt{\frac{x^{2} \cdot 1}{18}}$

Step 4.2

Factor the perfect power $3^{2}$ out of $18$ .

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

Step 4.3

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

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

Step 5

Pull terms out from under the radical.

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

Step 6

Rewrite using the commutative property of multiplication.

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

Step 7

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Step 8

Cancel the common factor of $\sqrt{2}$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

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

Step 9

Combine $\sqrt{1}$ and $\frac{1}{x}$ .

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

Step 10

Cancel the common factor of $x$ .

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Step 10.1

Cancel the common factor.

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

Step 10.2

Rewrite the expression.

$\sqrt{1} \frac{1}{3}$

Step 11

Combine $\sqrt{1}$ and $\frac{1}{3}$ .

$\frac{\sqrt{1}}{3}$

Step 12

Any root of $1$ is $1$ .

$\frac{1}{3}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{3}$

Decimal Form:

$0.\overline{3}$