Algebra Examples

$- \sqrt{4} + \sqrt{- 108} + \sqrt{9} - \sqrt{- 12}$

Step 1

Simplify each term.

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Step 1.1

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

$- \sqrt{2^{2}} + \sqrt{- 108} + \sqrt{9} - \sqrt{- 12}$

Step 1.2

Pull terms out from under the radical.

$- | 2 | + \sqrt{- 108} + \sqrt{9} - \sqrt{- 12}$

Step 1.3

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

$- 1 \cdot 2 + \sqrt{- 108} + \sqrt{9} - \sqrt{- 12}$

Step 1.4

Multiply $- 1$ by $2$ .

$- 2 + \sqrt{- 108} + \sqrt{9} - \sqrt{- 12}$

Step 1.5

Rewrite $- 108$ as $- 1 (108)$ .

$- 2 + \sqrt{- 1 (108)} + \sqrt{9} - \sqrt{- 12}$

Step 1.6

Rewrite $\sqrt{- 1 (108)}$ as $\sqrt{- 1} \cdot \sqrt{108}$ .

$- 2 + \sqrt{- 1} \cdot \sqrt{108} + \sqrt{9} - \sqrt{- 12}$

Step 1.7

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

$- 2 + i \cdot \sqrt{108} + \sqrt{9} - \sqrt{- 12}$

Step 1.8

Rewrite $108$ as $6^{2} \cdot 3$ .

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Step 1.8.1

Factor $36$ out of $108$ .

$- 2 + i \cdot \sqrt{36 (3)} + \sqrt{9} - \sqrt{- 12}$

Step 1.8.2

Rewrite $36$ as $6^{2}$ .

$- 2 + i \cdot \sqrt{6^{2} \cdot 3} + \sqrt{9} - \sqrt{- 12}$

Step 1.9

Pull terms out from under the radical.

$- 2 + i \cdot (| 6 | \sqrt{3}) + \sqrt{9} - \sqrt{- 12}$

Step 1.10

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$- 2 + i \cdot (6 \sqrt{3}) + \sqrt{9} - \sqrt{- 12}$

Step 1.11

Move $6$ to the left of $i$ .

$- 2 + 6 \cdot i \sqrt{3} + \sqrt{9} - \sqrt{- 12}$

Step 1.12

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

$- 2 + 6 i \sqrt{3} + \sqrt{3^{2}} - \sqrt{- 12}$

Step 1.13

Pull terms out from under the radical.

$- 2 + 6 i \sqrt{3} + | 3 | - \sqrt{- 12}$

Step 1.14

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

$- 2 + 6 i \sqrt{3} + 3 - \sqrt{- 12}$

Step 1.15

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

$- 2 + 6 i \sqrt{3} + 3 - \sqrt{- 1 (12)}$

Step 1.16

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

$- 2 + 6 i \sqrt{3} + 3 - (\sqrt{- 1} \cdot \sqrt{12})$

Step 1.17

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

$- 2 + 6 i \sqrt{3} + 3 - (i \cdot \sqrt{12})$

Step 1.18

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

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Step 1.18.1

Factor $4$ out of $12$ .

$- 2 + 6 i \sqrt{3} + 3 - (i \cdot \sqrt{4 (3)})$

Step 1.18.2

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

$- 2 + 6 i \sqrt{3} + 3 - (i \cdot \sqrt{2^{2} \cdot 3})$

Step 1.19

Pull terms out from under the radical.

$- 2 + 6 i \sqrt{3} + 3 - (i \cdot (| 2 | \sqrt{3}))$

Step 1.20

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

$- 2 + 6 i \sqrt{3} + 3 - (i \cdot (2 \sqrt{3}))$

Step 1.21

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

$- 2 + 6 i \sqrt{3} + 3 - (2 \cdot i \sqrt{3})$

Step 1.22

Multiply $2$ by $- 1$ .

$- 2 + 6 i \sqrt{3} + 3 - 2 i \sqrt{3}$

Step 2

Add $- 2$ and $3$ .

$6 i \sqrt{3} + 1 - 2 i \sqrt{3}$

Step 3

Subtract $2 i \sqrt{3}$ from $6 i \sqrt{3}$ .

$4 i \sqrt{3} + 1$

Step 4

Reorder $4 i \sqrt{3}$ and $1$ .

$1 + 4 i \sqrt{3}$