Pre-Algebra Examples

$- 5 \sqrt{\frac{71}{- 32}}$

Step 1

Move the negative in front of the fraction.

$- 5 \sqrt{- \frac{71}{32}}$

Step 2

Rewrite $- \frac{71}{32}$ as ${(\frac{i}{4})}^{2} \frac{71}{2}$ .

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Step 2.1

Rewrite $- 1$ as $i^{2}$ .

$- 5 \sqrt{i^{2} \frac{71}{32}}$

Step 2.2

Factor the perfect power $1^{2}$ out of $71$ .

$- 5 \sqrt{i^{2} \frac{1^{2} \cdot 71}{32}}$

Step 2.3

Factor the perfect power $4^{2}$ out of $32$ .

$- 5 \sqrt{i^{2} \frac{1^{2} \cdot 71}{4^{2} \cdot 2}}$

Step 2.4

Rearrange the fraction $\frac{1^{2} \cdot 71}{4^{2} \cdot 2}$ .

$- 5 \sqrt{i^{2} ({(\frac{1}{4})}^{2} \frac{71}{2})}$

Step 2.5

Rewrite $i^{2} {(\frac{1}{4})}^{2}$ as ${(\frac{i}{4})}^{2}$ .

$- 5 \sqrt{{(\frac{i}{4})}^{2} \frac{71}{2}}$

Step 3

Pull terms out from under the radical.

$- 5 (\frac{i}{4} \sqrt{\frac{71}{2}})$

Step 4

Rewrite $\sqrt{\frac{71}{2}}$ as $\frac{\sqrt{71}}{\sqrt{2}}$ .

$- 5 (\frac{i}{4} \cdot \frac{\sqrt{71}}{\sqrt{2}})$

Step 5

Multiply $\frac{\sqrt{71}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$- 5 (\frac{i}{4} (\frac{\sqrt{71}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}))$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{\sqrt{71}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$- 5 (\frac{i}{4} \cdot \frac{\sqrt{71} \sqrt{2}}{\sqrt{2} \sqrt{2}})$

Step 6.2

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

$- 5 (\frac{i}{4} \cdot \frac{\sqrt{71} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})$

Step 6.3

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

$- 5 (\frac{i}{4} \cdot \frac{\sqrt{71} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})$

Step 6.4

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

$- 5 (\frac{i}{4} \cdot \frac{\sqrt{71} \sqrt{2}}{{\sqrt{2}}^{1 + 1}})$

Step 6.5

Add $1$ and $1$ .

$- 5 (\frac{i}{4} \cdot \frac{\sqrt{71} \sqrt{2}}{{\sqrt{2}}^{2}})$

Step 6.6

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

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Step 6.6.1

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

$- 5 (\frac{i}{4} \cdot \frac{\sqrt{71} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})$

Step 6.6.2

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

$- 5 (\frac{i}{4} \cdot \frac{\sqrt{71} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})$

Step 6.6.3

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

$- 5 (\frac{i}{4} \cdot \frac{\sqrt{71} \sqrt{2}}{2^{\frac{2}{2}}})$

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

$- 5 (\frac{i}{4} \cdot \frac{\sqrt{71} \sqrt{2}}{2^{\frac{2}{2}}})$

Step 6.6.4.2

Rewrite the expression.

$- 5 (\frac{i}{4} \cdot \frac{\sqrt{71} \sqrt{2}}{2^{1}})$

Step 6.6.5

Evaluate the exponent.

$- 5 (\frac{i}{4} \cdot \frac{\sqrt{71} \sqrt{2}}{2})$

Step 7

Simplify the numerator.

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Step 7.1

Combine using the product rule for radicals.

$- 5 (\frac{i}{4} \cdot \frac{\sqrt{71 \cdot 2}}{2})$

Step 7.2

Multiply $71$ by $2$ .

$- 5 (\frac{i}{4} \cdot \frac{\sqrt{142}}{2})$

Step 8

Multiply $\frac{i}{4} \cdot \frac{\sqrt{142}}{2}$ .

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Step 8.1

Multiply $\frac{i}{4}$ by $\frac{\sqrt{142}}{2}$ .

$- 5 \frac{i \sqrt{142}}{4 \cdot 2}$

Step 8.2

Multiply $4$ by $2$ .

$- 5 \frac{i \sqrt{142}}{8}$

Step 9

Combine $- 5$ and $\frac{i \sqrt{142}}{8}$ .

$\frac{- 5 (i \sqrt{142})}{8}$

Step 10

Move the negative in front of the fraction.

$- \frac{5 i \sqrt{142}}{8}$