Simplify the Radical Expression square root of 100- square root of -2- square root of 9+ square root of -162

$\sqrt{100} - \sqrt{- 2} - \sqrt{9} + \sqrt{- 162}$

Step 1

Simplify each term.

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

$\sqrt{10^{2}} - \sqrt{- 2} - \sqrt{9} + \sqrt{- 162}$

Pull terms out from under the radical.

$| 10 | - \sqrt{- 2} - \sqrt{9} + \sqrt{- 162}$

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

$10 - \sqrt{- 2} - \sqrt{9} + \sqrt{- 162}$

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

$10 - \sqrt{- 1 (2)} - \sqrt{9} + \sqrt{- 162}$

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

$10 - (\sqrt{- 1} \cdot \sqrt{2}) - \sqrt{9} + \sqrt{- 162}$

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

$10 - (i \cdot \sqrt{2}) - \sqrt{9} + \sqrt{- 162}$

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

$10 - i \sqrt{2} - \sqrt{3^{2}} + \sqrt{- 162}$

Pull terms out from under the radical.

$10 - i \sqrt{2} - | 3 | + \sqrt{- 162}$

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

$10 - i \sqrt{2} - 1 \cdot 3 + \sqrt{- 162}$

Multiply $- 1$ by $3$ .

$10 - i \sqrt{2} - 3 + \sqrt{- 162}$

Rewrite $- 162$ as $- 1 (162)$ .

$10 - i \sqrt{2} - 3 + \sqrt{- 1 (162)}$

Rewrite $\sqrt{- 1 (162)}$ as $\sqrt{- 1} \cdot \sqrt{162}$ .

$10 - i \sqrt{2} - 3 + \sqrt{- 1} \cdot \sqrt{162}$

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

$10 - i \sqrt{2} - 3 + i \cdot \sqrt{162}$

Rewrite $162$ as $9^{2} \cdot 2$ .

Factor $81$ out of $162$ .

$10 - i \sqrt{2} - 3 + i \cdot \sqrt{81 (2)}$

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

$10 - i \sqrt{2} - 3 + i \cdot \sqrt{9^{2} \cdot 2}$

Pull terms out from under the radical.

$10 - i \sqrt{2} - 3 + i \cdot (| 9 | \sqrt{2})$

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

$10 - i \sqrt{2} - 3 + i \cdot (9 \sqrt{2})$

Move $9$ to the left of $i$ .

$10 - i \sqrt{2} - 3 + 9 i \sqrt{2}$

Step 2

Subtract $3$ from $10$ .

$- i \sqrt{2} + 7 + 9 i \sqrt{2}$

Step 3

Add $- i \sqrt{2}$ and $9 i \sqrt{2}$ .

$8 i \sqrt{2} + 7$

Step 4

Reorder $8 i \sqrt{2}$ and $7$ .

$7 + 8 i \sqrt{2}$

Simplify the Radical Expression square root of 100- square root of -2- square root of 9+ square root of -162

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