Simplify 2 square root of -180+3 square root of -125-4 square root of -20

$2 \sqrt{- 180} + 3 \sqrt{- 125} - 4 \sqrt{- 20}$

Step 1

Simplify each term.

Rewrite $- 180$ as $- 1 (180)$ .

$2 \sqrt{- 1 (180)} + 3 \sqrt{- 125} - 4 \sqrt{- 20}$

Rewrite $\sqrt{- 1 (180)}$ as $\sqrt{- 1} \cdot \sqrt{180}$ .

$2 (\sqrt{- 1} \cdot \sqrt{180}) + 3 \sqrt{- 125} - 4 \sqrt{- 20}$

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

$2 (i \cdot \sqrt{180}) + 3 \sqrt{- 125} - 4 \sqrt{- 20}$

Rewrite $180$ as $6^{2} \cdot 5$ .

Factor $36$ out of $180$ .

$2 (i \cdot \sqrt{36 (5)}) + 3 \sqrt{- 125} - 4 \sqrt{- 20}$

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

$2 (i \cdot \sqrt{6^{2} \cdot 5}) + 3 \sqrt{- 125} - 4 \sqrt{- 20}$

Pull terms out from under the radical.

$2 (i \cdot (6 \sqrt{5})) + 3 \sqrt{- 125} - 4 \sqrt{- 20}$

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

$2 (6 \cdot i \sqrt{5}) + 3 \sqrt{- 125} - 4 \sqrt{- 20}$

Multiply $6$ by $2$ .

$12 (i \sqrt{5}) + 3 \sqrt{- 125} - 4 \sqrt{- 20}$

Rewrite $- 125$ as $- 1 (125)$ .

$12 i \sqrt{5} + 3 \sqrt{- 1 (125)} - 4 \sqrt{- 20}$

Rewrite $\sqrt{- 1 (125)}$ as $\sqrt{- 1} \cdot \sqrt{125}$ .

$12 i \sqrt{5} + 3 (\sqrt{- 1} \cdot \sqrt{125}) - 4 \sqrt{- 20}$

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

$12 i \sqrt{5} + 3 (i \cdot \sqrt{125}) - 4 \sqrt{- 20}$

Rewrite $125$ as $5^{2} \cdot 5$ .

Factor $25$ out of $125$ .

$12 i \sqrt{5} + 3 (i \cdot \sqrt{25 (5)}) - 4 \sqrt{- 20}$

Rewrite $25$ as $5^{2}$ .

$12 i \sqrt{5} + 3 (i \cdot \sqrt{5^{2} \cdot 5}) - 4 \sqrt{- 20}$

Pull terms out from under the radical.

$12 i \sqrt{5} + 3 (i \cdot (5 \sqrt{5})) - 4 \sqrt{- 20}$

Move $5$ to the left of $i$ .

$12 i \sqrt{5} + 3 (5 \cdot i \sqrt{5}) - 4 \sqrt{- 20}$

Multiply $5$ by $3$ .

$12 i \sqrt{5} + 15 (i \sqrt{5}) - 4 \sqrt{- 20}$

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

$12 i \sqrt{5} + 15 i \sqrt{5} - 4 \sqrt{- 1 (20)}$

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

$12 i \sqrt{5} + 15 i \sqrt{5} - 4 (\sqrt{- 1} \cdot \sqrt{20})$

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

$12 i \sqrt{5} + 15 i \sqrt{5} - 4 (i \cdot \sqrt{20})$

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

Factor $4$ out of $20$ .

$12 i \sqrt{5} + 15 i \sqrt{5} - 4 (i \cdot \sqrt{4 (5)})$

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

$12 i \sqrt{5} + 15 i \sqrt{5} - 4 (i \cdot \sqrt{2^{2} \cdot 5})$

Pull terms out from under the radical.

$12 i \sqrt{5} + 15 i \sqrt{5} - 4 (i \cdot (2 \sqrt{5}))$

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

$12 i \sqrt{5} + 15 i \sqrt{5} - 4 (2 \cdot i \sqrt{5})$

Multiply $2$ by $- 4$ .

$12 i \sqrt{5} + 15 i \sqrt{5} - 8 i \sqrt{5}$

Step 2

Simplify by adding terms.

Add $12 i \sqrt{5}$ and $15 i \sqrt{5}$ .

$27 i \sqrt{5} - 8 i \sqrt{5}$

Subtract $8 i \sqrt{5}$ from $27 i \sqrt{5}$ .

$19 i \sqrt{5}$

Simplify 2 square root of -180+3 square root of -125-4 square root of -20

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