Evaluate ( square root of 12- square root of -20)-( square root of 48- square root of -80)

$(\sqrt{12} - \sqrt{- 20}) - (\sqrt{48} - \sqrt{- 80})$

Step 1

Simplify each term.

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

Factor $4$ out of $12$ .

$\sqrt{4 (3)} - \sqrt{- 20} - (\sqrt{48} - \sqrt{- 80})$

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

$\sqrt{2^{2} \cdot 3} - \sqrt{- 20} - (\sqrt{48} - \sqrt{- 80})$

Pull terms out from under the radical.

$2 \sqrt{3} - \sqrt{- 20} - (\sqrt{48} - \sqrt{- 80})$

Rewrite $- 20$ as $- 1 (20)$ .

$2 \sqrt{3} - \sqrt{- 1 (20)} - (\sqrt{48} - \sqrt{- 80})$

Rewrite $\sqrt{- 1 (20)}$ as $\sqrt{- 1} \cdot \sqrt{20}$ .

$2 \sqrt{3} - (\sqrt{- 1} \cdot \sqrt{20}) - (\sqrt{48} - \sqrt{- 80})$

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

$2 \sqrt{3} - (i \cdot \sqrt{20}) - (\sqrt{48} - \sqrt{- 80})$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$2 \sqrt{3} - (i \cdot \sqrt{4 (5)}) - (\sqrt{48} - \sqrt{- 80})$

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

$2 \sqrt{3} - (i \cdot \sqrt{2^{2} \cdot 5}) - (\sqrt{48} - \sqrt{- 80})$

Pull terms out from under the radical.

$2 \sqrt{3} - (i \cdot (2 \sqrt{5})) - (\sqrt{48} - \sqrt{- 80})$

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

$2 \sqrt{3} - (2 \cdot i \sqrt{5}) - (\sqrt{48} - \sqrt{- 80})$

Multiply $2$ by $- 1$ .

$2 \sqrt{3} - 2 (i \sqrt{5}) - (\sqrt{48} - \sqrt{- 80})$

Rewrite $- 80$ as $- 1 (80)$ .

$2 \sqrt{3} - 2 i \sqrt{5} - (\sqrt{48} - \sqrt{- 1 (80)})$

Rewrite $\sqrt{- 1 (80)}$ as $\sqrt{- 1} \cdot \sqrt{80}$ .

$2 \sqrt{3} - 2 i \sqrt{5} - (\sqrt{48} - (\sqrt{- 1} \cdot \sqrt{80}))$

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

$2 \sqrt{3} - 2 i \sqrt{5} - (\sqrt{48} - (i \cdot \sqrt{80}))$

Simplify each term.

Rewrite $48$ as $4^{2} \cdot 3$ .

Factor $16$ out of $48$ .

$2 \sqrt{3} - 2 i \sqrt{5} - (\sqrt{16 (3)} - (i \cdot \sqrt{80}))$

Rewrite $16$ as $4^{2}$ .

$2 \sqrt{3} - 2 i \sqrt{5} - (\sqrt{4^{2} \cdot 3} - (i \cdot \sqrt{80}))$

Pull terms out from under the radical.

$2 \sqrt{3} - 2 i \sqrt{5} - (4 \sqrt{3} - (i \cdot \sqrt{80}))$

Rewrite $80$ as $4^{2} \cdot 5$ .

Factor $16$ out of $80$ .

$2 \sqrt{3} - 2 i \sqrt{5} - (4 \sqrt{3} - (i \cdot \sqrt{16 (5)}))$

Rewrite $16$ as $4^{2}$ .

$2 \sqrt{3} - 2 i \sqrt{5} - (4 \sqrt{3} - (i \cdot \sqrt{4^{2} \cdot 5}))$

Pull terms out from under the radical.

$2 \sqrt{3} - 2 i \sqrt{5} - (4 \sqrt{3} - (i \cdot (4 \sqrt{5})))$

Move $4$ to the left of $i$ .

$2 \sqrt{3} - 2 i \sqrt{5} - (4 \sqrt{3} - (4 \cdot i \sqrt{5}))$

Multiply $4$ by $- 1$ .

$2 \sqrt{3} - 2 i \sqrt{5} - (4 \sqrt{3} - 4 i \sqrt{5})$

Apply the distributive property.

$2 \sqrt{3} - 2 i \sqrt{5} - (4 \sqrt{3}) - (- 4 i \sqrt{5})$

Multiply $4$ by $- 1$ .

$2 \sqrt{3} - 2 i \sqrt{5} - 4 \sqrt{3} - (- 4 i \sqrt{5})$

Multiply $- 4$ by $- 1$ .

$2 \sqrt{3} - 2 i \sqrt{5} - 4 \sqrt{3} + 4 i \sqrt{5}$

Step 2

Simplify by adding terms.

Subtract $4 \sqrt{3}$ from $2 \sqrt{3}$ .

$- 2 i \sqrt{5} - 2 \sqrt{3} + 4 i \sqrt{5}$

Add $- 2 i \sqrt{5}$ and $4 i \sqrt{5}$ .

$2 i \sqrt{5} - 2 \sqrt{3}$

Reorder $2 i \sqrt{5}$ and $- 2 \sqrt{3}$ .

$- 2 \sqrt{3} + 2 i \sqrt{5}$

Evaluate ( square root of 12- square root of -20)-( square root of 48- square root of -80)

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