Simplify ( square root of 18+ square root of -12)+(4 square root of 2- square root of -3)

$(\sqrt{18} + \sqrt{- 12}) + (4 \sqrt{2} - \sqrt{- 3})$

Step 1

Remove parentheses.

$\sqrt{18} + \sqrt{- 12} + 4 \sqrt{2} - \sqrt{- 3}$

Step 2

Simplify each term.

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$\sqrt{9 (2)} + \sqrt{- 12} + 4 \sqrt{2} - \sqrt{- 3}$

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

$\sqrt{3^{2} \cdot 2} + \sqrt{- 12} + 4 \sqrt{2} - \sqrt{- 3}$

Pull terms out from under the radical.

$3 \sqrt{2} + \sqrt{- 12} + 4 \sqrt{2} - \sqrt{- 3}$

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

$3 \sqrt{2} + \sqrt{- 1 (12)} + 4 \sqrt{2} - \sqrt{- 3}$

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

$3 \sqrt{2} + \sqrt{- 1} \cdot \sqrt{12} + 4 \sqrt{2} - \sqrt{- 3}$

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

$3 \sqrt{2} + i \cdot \sqrt{12} + 4 \sqrt{2} - \sqrt{- 3}$

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

Factor $4$ out of $12$ .

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

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

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

Pull terms out from under the radical.

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

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

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

Rewrite $- 3$ as $- 1 (3)$ .

$3 \sqrt{2} + 2 i \sqrt{3} + 4 \sqrt{2} - \sqrt{- 1 (3)}$

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$3 \sqrt{2} + 2 i \sqrt{3} + 4 \sqrt{2} - (\sqrt{- 1} \cdot \sqrt{3})$

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

$3 \sqrt{2} + 2 i \sqrt{3} + 4 \sqrt{2} - i \sqrt{3}$

Step 3

Simplify by adding terms.

Add $3 \sqrt{2}$ and $4 \sqrt{2}$ .

$2 i \sqrt{3} + 7 \sqrt{2} - i \sqrt{3}$

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

$1 i \sqrt{3} + 7 \sqrt{2}$

Simplify the expression.

Multiply $i$ by $1$ .

$i \sqrt{3} + 7 \sqrt{2}$

Reorder $i \sqrt{3}$ and $7 \sqrt{2}$ .

$7 \sqrt{2} + i \sqrt{3}$