Pre-Algebra Examples

$\sqrt{\frac{12}{x^{4}}} - \sqrt{\frac{3}{25 x^{4}}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\frac{12}{x^{4}}$ as ${(\frac{2}{x^{2}})}^{2} \cdot 3$ .

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Step 1.1.1

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

$\sqrt{\frac{2^{2} \cdot 3}{x^{4}}} - \sqrt{\frac{3}{25 x^{4}}}$

Step 1.1.2

Factor the perfect power ${(x^{2})}^{2}$ out of $x^{4}$ .

$\sqrt{\frac{2^{2} \cdot 3}{{(x^{2})}^{2} \cdot 1}} - \sqrt{\frac{3}{25 x^{4}}}$

Step 1.1.3

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

$\sqrt{{(\frac{2}{x^{2}})}^{2} \cdot 3} - \sqrt{\frac{3}{25 x^{4}}}$

Step 1.2

Pull terms out from under the radical.

$\frac{2}{x^{2}} \sqrt{3} - \sqrt{\frac{3}{25 x^{4}}}$

Step 1.3

Combine $\frac{2}{x^{2}}$ and $\sqrt{3}$ .

$\frac{2 \sqrt{3}}{x^{2}} - \sqrt{\frac{3}{25 x^{4}}}$

Step 1.4

Rewrite $\frac{3}{25 x^{4}}$ as ${(\frac{1}{5 x^{2}})}^{2} \cdot 3$ .

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Step 1.4.1

Factor the perfect power $1^{2}$ out of $3$ .

$\frac{2 \sqrt{3}}{x^{2}} - \sqrt{\frac{1^{2} \cdot 3}{25 x^{4}}}$

Step 1.4.2

Factor the perfect power ${(5 x^{2})}^{2}$ out of $25 x^{4}$ .

$\frac{2 \sqrt{3}}{x^{2}} - \sqrt{\frac{1^{2} \cdot 3}{{(5 x^{2})}^{2} \cdot 1}}$

Step 1.4.3

Rearrange the fraction $\frac{1^{2} \cdot 3}{{(5 x^{2})}^{2} \cdot 1}$ .

$\frac{2 \sqrt{3}}{x^{2}} - \sqrt{{(\frac{1}{5 x^{2}})}^{2} \cdot 3}$

Step 1.5

Pull terms out from under the radical.

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

Step 1.6

Combine $\frac{1}{5 x^{2}}$ and $\sqrt{3}$ .

$\frac{2 \sqrt{3}}{x^{2}} - \frac{\sqrt{3}}{5 x^{2}}$

Step 2

To write $\frac{2 \sqrt{3}}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{2 \sqrt{3}}{x^{2}} \cdot \frac{5}{5} - \frac{\sqrt{3}}{5 x^{2}}$

Step 3

Write each expression with a common denominator of $5 x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

$\frac{2 \sqrt{3} \cdot 5}{x^{2} \cdot 5} - \frac{\sqrt{3}}{5 x^{2}}$

Step 3.2

Move $5$ to the left of $x^{2}$ .

$\frac{2 \sqrt{3} \cdot 5}{5 x^{2}} - \frac{\sqrt{3}}{5 x^{2}}$

Step 4

Combine the numerators over the common denominator.

$\frac{2 \sqrt{3} \cdot 5 - \sqrt{3}}{5 x^{2}}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $5$ by $2$ .

$\frac{10 \sqrt{3} - \sqrt{3}}{5 x^{2}}$

Step 5.2

Subtract $\sqrt{3}$ from $10 \sqrt{3}$ .

$\frac{9 \sqrt{3}}{5 x^{2}}$