Algebra Examples

$\frac{- 8 + \sqrt{- 12}}{40}$

Step 1

Pull out imaginary unit $i$ .

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Step 1.1

Rewrite $- 12$ as $- 1 (12)$ .

$\frac{- 8 + \sqrt{- 1 (12)}}{40}$

Step 1.2

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$\frac{- 8 + \sqrt{- 1} \cdot \sqrt{12}}{40}$

Step 1.3

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

$\frac{- 8 + i \cdot \sqrt{12}}{40}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.1.1

Factor $4$ out of $12$ .

$\frac{- 8 + i \cdot \sqrt{4 (3)}}{40}$

Step 2.1.2

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

$\frac{- 8 + i \cdot \sqrt{2^{2} \cdot 3}}{40}$

Step 2.2

Pull terms out from under the radical.

$\frac{- 8 + i \cdot (2 \sqrt{3})}{40}$

Step 2.3

Move $2$ to the left of $i$ .

$\frac{- 8 + 2 i \sqrt{3}}{40}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $- 8 + 2 i \sqrt{3}$ and $40$ .

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Step 3.1.1

Factor $2$ out of $- 8$ .

$\frac{2 \cdot - 4 + 2 i \sqrt{3}}{40}$

Step 3.1.2

Factor $2$ out of $2 i \sqrt{3}$ .

$\frac{2 \cdot - 4 + 2 (i \sqrt{3})}{40}$

Step 3.1.3

Factor $2$ out of $2 \cdot - 4 + 2 (i \sqrt{3})$ .

$\frac{2 \cdot (- 4 + i \sqrt{3})}{40}$

Step 3.1.4

Cancel the common factors.

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Step 3.1.4.1

Factor $2$ out of $40$ .

$\frac{2 \cdot (- 4 + i \sqrt{3})}{2 (20)}$

Step 3.1.4.2

Cancel the common factor.

$\frac{2 \cdot (- 4 + i \sqrt{3})}{2 \cdot 20}$

Step 3.1.4.3

Rewrite the expression.

$\frac{- 4 + i \sqrt{3}}{20}$

Step 3.2

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

$\frac{- 1 (4) + i \sqrt{3}}{20}$

Step 3.3

Factor $- 1$ out of $i \sqrt{3}$ .

$\frac{- 1 (4) - (- i \sqrt{3})}{20}$

Step 3.4

Factor $- 1$ out of $- 1 (4) - (- i \sqrt{3})$ .

$\frac{- 1 (4 - i \sqrt{3})}{20}$

Step 3.5

Move the negative in front of the fraction.

$- \frac{4 - i \sqrt{3}}{20}$