Simplify - square root of 121- square root of -160+ square root of 16- square root of -10

$- \sqrt{121} - \sqrt{- 160} + \sqrt{16} - \sqrt{- 10}$

Step 1

Simplify each term.

Rewrite $121$ as $11^{2}$ .

$- \sqrt{11^{2}} - \sqrt{- 160} + \sqrt{16} - \sqrt{- 10}$

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

$- 1 \cdot 11 - \sqrt{- 160} + \sqrt{16} - \sqrt{- 10}$

Multiply $- 1$ by $11$ .

$- 11 - \sqrt{- 160} + \sqrt{16} - \sqrt{- 10}$

Rewrite $- 160$ as $- 1 (160)$ .

$- 11 - \sqrt{- 1 (160)} + \sqrt{16} - \sqrt{- 10}$

Rewrite $\sqrt{- 1 (160)}$ as $\sqrt{- 1} \cdot \sqrt{160}$ .

$- 11 - (\sqrt{- 1} \cdot \sqrt{160}) + \sqrt{16} - \sqrt{- 10}$

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

$- 11 - (i \cdot \sqrt{160}) + \sqrt{16} - \sqrt{- 10}$

Rewrite $160$ as $4^{2} \cdot 10$ .

Factor $16$ out of $160$ .

$- 11 - (i \cdot \sqrt{16 (10)}) + \sqrt{16} - \sqrt{- 10}$

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

$- 11 - (i \cdot \sqrt{4^{2} \cdot 10}) + \sqrt{16} - \sqrt{- 10}$

Pull terms out from under the radical.

$- 11 - (i \cdot (4 \sqrt{10})) + \sqrt{16} - \sqrt{- 10}$

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

$- 11 - (4 \cdot i \sqrt{10}) + \sqrt{16} - \sqrt{- 10}$

Multiply $4$ by $- 1$ .

$- 11 - 4 (i \sqrt{10}) + \sqrt{16} - \sqrt{- 10}$

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

$- 11 - 4 i \sqrt{10} + \sqrt{4^{2}} - \sqrt{- 10}$

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

$- 11 - 4 i \sqrt{10} + 4 - \sqrt{- 10}$

Rewrite $- 10$ as $- 1 (10)$ .

$- 11 - 4 i \sqrt{10} + 4 - \sqrt{- 1 (10)}$

Rewrite $\sqrt{- 1 (10)}$ as $\sqrt{- 1} \cdot \sqrt{10}$ .

$- 11 - 4 i \sqrt{10} + 4 - (\sqrt{- 1} \cdot \sqrt{10})$

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

$- 11 - 4 i \sqrt{10} + 4 - i \sqrt{10}$

Step 2

Simplify by adding terms.

Add $- 11$ and $4$ .

$- 4 i \sqrt{10} - 7 - i \sqrt{10}$

Subtract $i \sqrt{10}$ from $- 4 i \sqrt{10}$ .

$- 5 i \sqrt{10} - 7$

Reorder $- 5 i \sqrt{10}$ and $- 7$ .

$- 7 - 5 i \sqrt{10}$