Simplify square root of -16+ square root of 8+ square root of -36

$\sqrt{- 16} + \sqrt{8} + \sqrt{- 36}$

Step 1

Simplify each term.

Rewrite $- 16$ as $- 1 (16)$ .

$\sqrt{- 1 (16)} + \sqrt{8} + \sqrt{- 36}$

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$\sqrt{- 1} \cdot \sqrt{16} + \sqrt{8} + \sqrt{- 36}$

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

$i \cdot \sqrt{16} + \sqrt{8} + \sqrt{- 36}$

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

$i \cdot \sqrt{4^{2}} + \sqrt{8} + \sqrt{- 36}$

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

$i \cdot 4 + \sqrt{8} + \sqrt{- 36}$

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

$4 \cdot i + \sqrt{8} + \sqrt{- 36}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$4 i + \sqrt{4 (2)} + \sqrt{- 36}$

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

$4 i + \sqrt{2^{2} \cdot 2} + \sqrt{- 36}$

Pull terms out from under the radical.

$4 i + 2 \sqrt{2} + \sqrt{- 36}$

Rewrite $- 36$ as $- 1 (36)$ .

$4 i + 2 \sqrt{2} + \sqrt{- 1 (36)}$

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$4 i + 2 \sqrt{2} + \sqrt{- 1} \cdot \sqrt{36}$

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

$4 i + 2 \sqrt{2} + i \cdot \sqrt{36}$

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

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

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

$4 i + 2 \sqrt{2} + i \cdot 6$

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

$4 i + 2 \sqrt{2} + 6 i$

Step 2

Simplify by adding terms.

Add $4 i$ and $6 i$ .

$10 i + 2 \sqrt{2}$

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

$2 \sqrt{2} + 10 i$