Simplify the Radical Expression square root of 144+ square root of -8+ square root of 100- square root of -2

$\sqrt{144} + \sqrt{- 8} + \sqrt{100} - \sqrt{- 2}$

Step 1

Simplify each term.

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

$\sqrt{12^{2}} + \sqrt{- 8} + \sqrt{100} - \sqrt{- 2}$

Pull terms out from under the radical.

$| 12 | + \sqrt{- 8} + \sqrt{100} - \sqrt{- 2}$

The absolute value is the distance between a number and zero. The distance between $0$ and $12$ is $12$ .

$12 + \sqrt{- 8} + \sqrt{100} - \sqrt{- 2}$

Rewrite $- 8$ as $- 1 (8)$ .

$12 + \sqrt{- 1 (8)} + \sqrt{100} - \sqrt{- 2}$

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$12 + \sqrt{- 1} \cdot \sqrt{8} + \sqrt{100} - \sqrt{- 2}$

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

$12 + i \cdot \sqrt{8} + \sqrt{100} - \sqrt{- 2}$

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

Factor $4$ out of $8$ .

$12 + i \cdot \sqrt{4 (2)} + \sqrt{100} - \sqrt{- 2}$

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

$12 + i \cdot \sqrt{2^{2} \cdot 2} + \sqrt{100} - \sqrt{- 2}$

Pull terms out from under the radical.

$12 + i \cdot (| 2 | \sqrt{2}) + \sqrt{100} - \sqrt{- 2}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$12 + i \cdot (2 \sqrt{2}) + \sqrt{100} - \sqrt{- 2}$

Move $2$ to the left of $i$ .

$12 + 2 \cdot i \sqrt{2} + \sqrt{100} - \sqrt{- 2}$

Rewrite $100$ as $10^{2}$ .

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

Pull terms out from under the radical.

$12 + 2 i \sqrt{2} + | 10 | - \sqrt{- 2}$

The absolute value is the distance between a number and zero. The distance between $0$ and $10$ is $10$ .

$12 + 2 i \sqrt{2} + 10 - \sqrt{- 2}$

Rewrite $- 2$ as $- 1 (2)$ .

$12 + 2 i \sqrt{2} + 10 - \sqrt{- 1 (2)}$

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$12 + 2 i \sqrt{2} + 10 - (\sqrt{- 1} \cdot \sqrt{2})$

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

$12 + 2 i \sqrt{2} + 10 - i \sqrt{2}$

Step 2

Add $12$ and $10$ .

$2 i \sqrt{2} + 22 - i \sqrt{2}$

Step 3

Subtract $i \sqrt{2}$ from $2 i \sqrt{2}$ .

$1 i \sqrt{2} + 22$

Step 4

Multiply $i$ by $1$ .

$i \sqrt{2} + 22$

Step 5

Reorder $i \sqrt{2}$ and $22$ .

$22 + i \sqrt{2}$

Simplify the Radical Expression square root of 144+ square root of -8+ square root of 100- square root of -2

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