Algebra Examples

$\frac{- 3 + \sqrt{- 27}}{3}$

Step 1

Pull out imaginary unit $i$ .

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Step 1.1

Rewrite $- 27$ as $- 1 (27)$ .

$\frac{- 3 + \sqrt{- 1 (27)}}{3}$

Step 1.2

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$\frac{- 3 + \sqrt{- 1} \cdot \sqrt{27}}{3}$

Step 1.3

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

$\frac{- 3 + i \cdot \sqrt{27}}{3}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 2.1.1

Factor $9$ out of $27$ .

$\frac{- 3 + i \cdot \sqrt{9 (3)}}{3}$

Step 2.1.2

Rewrite $9$ as $3^{2}$ .

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

Step 2.2

Pull terms out from under the radical.

$\frac{- 3 + i \cdot (| 3 | \sqrt{3})}{3}$

Step 2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

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

Step 2.4

Move $3$ to the left of $i$ .

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

Step 3

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

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Step 3.1

Factor $3$ out of $- 3$ .

$\frac{3 \cdot - 1 + 3 i \sqrt{3}}{3}$

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $3$ out of $3$ .

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

$\frac{- 1 + i \sqrt{3}}{1}$

Step 3.4.4

Divide $- 1 + i \sqrt{3}$ by $1$ .

$- 1 + i \sqrt{3}$