Algebra Examples

$\frac{- 2 + \sqrt{- 8}}{2}$

Step 1

Pull out imaginary unit $i$ .

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Step 1.1

Rewrite $- 8$ as $- 1 (8)$ .

$\frac{- 2 + \sqrt{- 1 (8)}}{2}$

Step 1.2

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$\frac{- 2 + \sqrt{- 1} \cdot \sqrt{8}}{2}$

Step 1.3

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

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

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.1.1

Factor $4$ out of $8$ .

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

Step 2.1.2

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

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

Step 2.2

Pull terms out from under the radical.

$\frac{- 2 + i \cdot (| 2 | \sqrt{2})}{2}$

Step 2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Step 2.4

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

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

Step 3

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

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Step 3.1

Factor $2$ out of $- 2$ .

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

Step 3.2

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

$\frac{2 \cdot - 1 + 2 (i \sqrt{2})}{2}$

Step 3.3

Factor $2$ out of $2 (- 1) + 2 (i \sqrt{2})$ .

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

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $2$ out of $2$ .

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

Step 3.4.2

Cancel the common factor.

$\frac{2 (- 1 + i \sqrt{2})}{2 \cdot 1}$

Step 3.4.3

Rewrite the expression.

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

Step 3.4.4

Divide $- 1 + i \sqrt{2}$ by $1$ .

$- 1 + i \sqrt{2}$