Pre-Algebra Examples

$\frac{\sqrt{- 9}}{\sqrt{- 4} \sqrt{- 36}}$

Step 1

Pull out imaginary unit $i$ .

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Step 1.1

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{\sqrt{- 1 (9)}}{\sqrt{- 4} \sqrt{- 36}}$

Step 1.2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$\frac{\sqrt{- 1} \cdot \sqrt{9}}{\sqrt{- 4} \sqrt{- 36}}$

Step 1.3

Rewrite $\sqrt{- 1}$ as $i$ .

$\frac{i \cdot \sqrt{9}}{\sqrt{- 4} \sqrt{- 36}}$

Step 1.4

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{i \cdot \sqrt{9}}{\sqrt{- 1 (4)} \sqrt{- 36}}$

Step 1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$\frac{i \cdot \sqrt{9}}{\sqrt{- 1} \cdot \sqrt{4} \sqrt{- 36}}$

Step 1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$\frac{i \cdot \sqrt{9}}{i \cdot \sqrt{4} \sqrt{- 36}}$

Step 1.7

Rewrite $- 36$ as $- 1 (36)$ .

$\frac{i \cdot \sqrt{9}}{i \cdot \sqrt{4} \sqrt{- 1 (36)}}$

Step 1.8

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$\frac{i \cdot \sqrt{9}}{i \cdot \sqrt{4} (\sqrt{- 1} \cdot \sqrt{36})}$

Step 1.9

Rewrite $\sqrt{- 1}$ as $i$ .

$\frac{i \cdot \sqrt{9}}{i \cdot \sqrt{4} (i \cdot \sqrt{36})}$

Step 2

Combine $\sqrt{9}$ and $\sqrt{36}$ into a single radical.

$\frac{i \cdot \sqrt{\frac{9}{36}}}{i \cdot \sqrt{4} i}$

Step 3

Cancel the common factor of $9$ and $36$ .

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Step 3.1

Factor $9$ out of $9$ .

$\frac{i \cdot \sqrt{\frac{9 (1)}{36}}}{i \cdot \sqrt{4} i}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $9$ out of $36$ .

$\frac{i \cdot \sqrt{\frac{9 \cdot 1}{9 \cdot 4}}}{i \cdot \sqrt{4} i}$

Step 3.2.2

Cancel the common factor.

$\frac{i \cdot \sqrt{\frac{9 \cdot 1}{9 \cdot 4}}}{i \cdot \sqrt{4} i}$

Step 3.2.3

Rewrite the expression.

$\frac{i \cdot \sqrt{\frac{1}{4}}}{i \cdot \sqrt{4} i}$

Step 4

Group $i$ and $i$ together.

$\frac{i \cdot \sqrt{\frac{1}{4}}}{i i \cdot \sqrt{4}}$

Step 5

Multiply $i i$ .

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Step 5.1

Raise $i$ to the power of $1$ .

$\frac{i \cdot \sqrt{\frac{1}{4}}}{i^{1} i \cdot \sqrt{4}}$

Step 5.2

Raise $i$ to the power of $1$ .

$\frac{i \cdot \sqrt{\frac{1}{4}}}{i^{1} i^{1} \cdot \sqrt{4}}$

Step 5.3

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

$\frac{i \cdot \sqrt{\frac{1}{4}}}{i^{1 + 1} \cdot \sqrt{4}}$

Step 5.4

Add $1$ and $1$ .

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

Step 6

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

$\frac{i \cdot \sqrt{\frac{1}{4}}}{- 1 \cdot \sqrt{4}}$

Step 7

Simplify the numerator.

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Step 7.1

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$\frac{i \frac{\sqrt{1}}{\sqrt{4}}}{- 1 \cdot \sqrt{4}}$

Step 7.2

Any root of $1$ is $1$ .

$\frac{i \frac{1}{\sqrt{4}}}{- 1 \cdot \sqrt{4}}$

Step 7.3

Simplify the denominator.

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Step 7.3.1

Rewrite $4$ as $2^{2}$ .

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

Step 7.3.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 8

Simplify the denominator.

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Step 8.1

Rewrite $4$ as $2^{2}$ .

$\frac{i \frac{1}{2}}{- 1 \sqrt{2^{2}}}$

Step 8.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{i \frac{1}{2}}{- 1 \cdot 2}$

Step 9

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

$\frac{\frac{i}{2}}{- 1 \cdot 2}$

Step 10

Multiply $- 1$ by $2$ .

$\frac{\frac{i}{2}}{- 2}$

Step 11

Multiply the numerator by the reciprocal of the denominator.

$\frac{i}{2} \cdot \frac{1}{- 2}$

Step 12

Move the negative in front of the fraction.

$\frac{i}{2} (- \frac{1}{2})$

Step 13

Multiply $\frac{i}{2} (- \frac{1}{2})$ .

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Step 13.1

Multiply $\frac{i}{2}$ by $\frac{1}{2}$ .

$- \frac{i}{2 \cdot 2}$

Step 13.2

Multiply $2$ by $2$ .

$- \frac{i}{4}$