Simplify - square root of 16- square root of -5- square root of 9+ square root of -180

$- \sqrt{16} - \sqrt{- 5} - \sqrt{9} + \sqrt{- 180}$

Step 1

Simplify each term.

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

$- \sqrt{4^{2}} - \sqrt{- 5} - \sqrt{9} + \sqrt{- 180}$

Pull terms out from under the radical, assuming positive real numbers.

$- 1 \cdot 4 - \sqrt{- 5} - \sqrt{9} + \sqrt{- 180}$

Multiply $- 1$ by $4$ .

$- 4 - \sqrt{- 5} - \sqrt{9} + \sqrt{- 180}$

Rewrite $- 5$ as $- 1 (5)$ .

$- 4 - \sqrt{- 1 (5)} - \sqrt{9} + \sqrt{- 180}$

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

$- 4 - (\sqrt{- 1} \cdot \sqrt{5}) - \sqrt{9} + \sqrt{- 180}$

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

$- 4 - (i \cdot \sqrt{5}) - \sqrt{9} + \sqrt{- 180}$

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

$- 4 - i \sqrt{5} - \sqrt{3^{2}} + \sqrt{- 180}$

Pull terms out from under the radical, assuming positive real numbers.

$- 4 - i \sqrt{5} - 1 \cdot 3 + \sqrt{- 180}$

Multiply $- 1$ by $3$ .

$- 4 - i \sqrt{5} - 3 + \sqrt{- 180}$

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

$- 4 - i \sqrt{5} - 3 + \sqrt{- 1 (180)}$

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

$- 4 - i \sqrt{5} - 3 + \sqrt{- 1} \cdot \sqrt{180}$

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

$- 4 - i \sqrt{5} - 3 + i \cdot \sqrt{180}$

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

Factor $36$ out of $180$ .

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

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

$- 4 - i \sqrt{5} - 3 + i \cdot \sqrt{6^{2} \cdot 5}$

Pull terms out from under the radical.

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

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

$- 4 - i \sqrt{5} - 3 + 6 i \sqrt{5}$

Step 2

Simplify by adding terms.

Subtract $3$ from $- 4$ .

$- i \sqrt{5} - 7 + 6 i \sqrt{5}$

Add $- i \sqrt{5}$ and $6 i \sqrt{5}$ .

$5 i \sqrt{5} - 7$

Reorder $5 i \sqrt{5}$ and $- 7$ .

$- 7 + 5 i \sqrt{5}$