Simplify square root of 144- square root of -90+ square root of 16+ square root of -10

$\sqrt{144} - \sqrt{- 90} + \sqrt{16} + \sqrt{- 10}$

Step 1

Simplify each term.

Rewrite $144$ as $12^{2}$ .

$\sqrt{12^{2}} - \sqrt{- 90} + \sqrt{16} + \sqrt{- 10}$

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

$12 - \sqrt{- 90} + \sqrt{16} + \sqrt{- 10}$

Rewrite $- 90$ as $- 1 (90)$ .

$12 - \sqrt{- 1 (90)} + \sqrt{16} + \sqrt{- 10}$

Rewrite $\sqrt{- 1 (90)}$ as $\sqrt{- 1} \cdot \sqrt{90}$ .

$12 - (\sqrt{- 1} \cdot \sqrt{90}) + \sqrt{16} + \sqrt{- 10}$

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

$12 - (i \cdot \sqrt{90}) + \sqrt{16} + \sqrt{- 10}$

Rewrite $90$ as $3^{2} \cdot 10$ .

Factor $9$ out of $90$ .

$12 - (i \cdot \sqrt{9 (10)}) + \sqrt{16} + \sqrt{- 10}$

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

$12 - (i \cdot \sqrt{3^{2} \cdot 10}) + \sqrt{16} + \sqrt{- 10}$

Pull terms out from under the radical.

$12 - (i \cdot (3 \sqrt{10})) + \sqrt{16} + \sqrt{- 10}$

Move $3$ to the left of $i$ .

$12 - (3 \cdot i \sqrt{10}) + \sqrt{16} + \sqrt{- 10}$

Multiply $3$ by $- 1$ .

$12 - 3 (i \sqrt{10}) + \sqrt{16} + \sqrt{- 10}$

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

$12 - 3 i \sqrt{10} + \sqrt{4^{2}} + \sqrt{- 10}$

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

$12 - 3 i \sqrt{10} + 4 + \sqrt{- 10}$

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

$12 - 3 i \sqrt{10} + 4 + \sqrt{- 1 (10)}$

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

$12 - 3 i \sqrt{10} + 4 + \sqrt{- 1} \cdot \sqrt{10}$

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

$12 - 3 i \sqrt{10} + 4 + i \sqrt{10}$

Step 2

Simplify by adding terms.

Add $12$ and $4$ .

$- 3 i \sqrt{10} + 16 + i \sqrt{10}$

Add $- 3 i \sqrt{10}$ and $i \sqrt{10}$ .

$- 2 i \sqrt{10} + 16$

Reorder $- 2 i \sqrt{10}$ and $16$ .

$16 - 2 i \sqrt{10}$