Simplify the Radical Expression - square root of 4+ square root of -108+ square root of 9- square root of -12

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

Step 1

Simplify each term.

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

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

Pull terms out from under the radical.

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

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

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

Multiply $- 1$ by $2$ .

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

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

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

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

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

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

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

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

Factor $36$ out of $108$ .

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

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

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

Pull terms out from under the radical.

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

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

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

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

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

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

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

Pull terms out from under the radical.

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

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

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

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

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

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

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

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

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

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

Factor $4$ out of $12$ .

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

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

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

Pull terms out from under the radical.

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

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

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

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

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

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 $1$ .

$1 + 4 i \sqrt{3}$

Simplify the Radical Expression - square root of 4+ square root of -108+ square root of 9- square root of -12

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