Simplify square root of 144+ square root of -5- square root of 1+ square root of -45

$\sqrt{144} + \sqrt{- 5} - \sqrt{1} + \sqrt{- 45}$

Step 1

Simplify each term.

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

$\sqrt{12^{2}} + \sqrt{- 5} - \sqrt{1} + \sqrt{- 45}$

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

$12 + \sqrt{- 5} - \sqrt{1} + \sqrt{- 45}$

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

$12 + \sqrt{- 1 (5)} - \sqrt{1} + \sqrt{- 45}$

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

$12 + \sqrt{- 1} \cdot \sqrt{5} - \sqrt{1} + \sqrt{- 45}$

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

$12 + i \cdot \sqrt{5} - \sqrt{1} + \sqrt{- 45}$

Any root of $1$ is $1$ .

$12 + i \sqrt{5} - 1 \cdot 1 + \sqrt{- 45}$

Multiply $- 1$ by $1$ .

$12 + i \sqrt{5} - 1 + \sqrt{- 45}$

Rewrite $- 45$ as $- 1 (45)$ .

$12 + i \sqrt{5} - 1 + \sqrt{- 1 (45)}$

Rewrite $\sqrt{- 1 (45)}$ as $\sqrt{- 1} \cdot \sqrt{45}$ .

$12 + i \sqrt{5} - 1 + \sqrt{- 1} \cdot \sqrt{45}$

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

$12 + i \sqrt{5} - 1 + i \cdot \sqrt{45}$

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$12 + i \sqrt{5} - 1 + i \cdot \sqrt{9 (5)}$

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

$12 + i \sqrt{5} - 1 + i \cdot \sqrt{3^{2} \cdot 5}$

Pull terms out from under the radical.

$12 + i \sqrt{5} - 1 + i \cdot (3 \sqrt{5})$

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

$12 + i \sqrt{5} - 1 + 3 i \sqrt{5}$

Step 2

Simplify by adding terms.

Subtract $1$ from $12$ .

$i \sqrt{5} + 11 + 3 i \sqrt{5}$

Add $i \sqrt{5}$ and $3 i \sqrt{5}$ .

$4 i \sqrt{5} + 11$

Reorder $4 i \sqrt{5}$ and $11$ .

$11 + 4 i \sqrt{5}$