Simplify the Radical Expression square root of 81+ square root of -7+ square root of 9+ square root of -112

$\sqrt{81} + \sqrt{- 7} + \sqrt{9} + \sqrt{- 112}$

Simplify each term.

Rewrite $81$ as $9^{2}$ .

$\sqrt{9^{2}} + \sqrt{- 7} + \sqrt{9} + \sqrt{- 112}$

Pull terms out from under the radical.

$| 9 | + \sqrt{- 7} + \sqrt{9} + \sqrt{- 112}$

The absolute value is the distance between a number and zero. The distance between $0$ and $9$ is $9$ .

$9 + \sqrt{- 7} + \sqrt{9} + \sqrt{- 112}$

Rewrite $- 7$ as $- 1 (7)$ .

$9 + \sqrt{- 1 (7)} + \sqrt{9} + \sqrt{- 112}$

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$9 + \sqrt{- 1} \cdot \sqrt{7} + \sqrt{9} + \sqrt{- 112}$

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

$9 + i \cdot \sqrt{7} + \sqrt{9} + \sqrt{- 112}$

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

$9 + i \sqrt{7} + \sqrt{3^{2}} + \sqrt{- 112}$

Pull terms out from under the radical.

$9 + i \sqrt{7} + | 3 | + \sqrt{- 112}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$9 + i \sqrt{7} + 3 + \sqrt{- 112}$

Rewrite $- 112$ as $- 1 (112)$ .

$9 + i \sqrt{7} + 3 + \sqrt{- 1 (112)}$

Rewrite $\sqrt{- 1 (112)}$ as $\sqrt{- 1} \cdot \sqrt{112}$ .

$9 + i \sqrt{7} + 3 + \sqrt{- 1} \cdot \sqrt{112}$

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

$9 + i \sqrt{7} + 3 + i \cdot \sqrt{112}$

Rewrite $112$ as $4^{2} \cdot 7$ .

Factor $16$ out of $112$ .

$9 + i \sqrt{7} + 3 + i \cdot \sqrt{16 (7)}$

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

$9 + i \sqrt{7} + 3 + i \cdot \sqrt{4^{2} \cdot 7}$

Pull terms out from under the radical.

$9 + i \sqrt{7} + 3 + i \cdot (| 4 | \sqrt{7})$

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$9 + i \sqrt{7} + 3 + i \cdot (4 \sqrt{7})$

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

$9 + i \sqrt{7} + 3 + 4 i \sqrt{7}$

Add $9$ and $3$ .

$i \sqrt{7} + 12 + 4 i \sqrt{7}$

Add $i \sqrt{7}$ and $4 i \sqrt{7}$ .

$5 i \sqrt{7} + 12$

Reorder $5 i \sqrt{7}$ and $12$ .

$12 + 5 i \sqrt{7}$