Simplify the Radical Expression - square root of 36+ square root of -72- square root of 1+ square root of -162

$- \sqrt{36} + \sqrt{- 72} - \sqrt{1} + \sqrt{- 162}$

Step 1

Simplify each term.

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

$- \sqrt{6^{2}} + \sqrt{- 72} - \sqrt{1} + \sqrt{- 162}$

Pull terms out from under the radical.

$- | 6 | + \sqrt{- 72} - \sqrt{1} + \sqrt{- 162}$

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$- 1 \cdot 6 + \sqrt{- 72} - \sqrt{1} + \sqrt{- 162}$

Multiply $- 1$ by $6$ .

$- 6 + \sqrt{- 72} - \sqrt{1} + \sqrt{- 162}$

Rewrite $- 72$ as $- 1 (72)$ .

$- 6 + \sqrt{- 1 (72)} - \sqrt{1} + \sqrt{- 162}$

Rewrite $\sqrt{- 1 (72)}$ as $\sqrt{- 1} \cdot \sqrt{72}$ .

$- 6 + \sqrt{- 1} \cdot \sqrt{72} - \sqrt{1} + \sqrt{- 162}$

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

$- 6 + i \cdot \sqrt{72} - \sqrt{1} + \sqrt{- 162}$

Rewrite $72$ as $6^{2} \cdot 2$ .

Factor $36$ out of $72$ .

$- 6 + i \cdot \sqrt{36 (2)} - \sqrt{1} + \sqrt{- 162}$

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

$- 6 + i \cdot \sqrt{6^{2} \cdot 2} - \sqrt{1} + \sqrt{- 162}$

Pull terms out from under the radical.

$- 6 + i \cdot (| 6 | \sqrt{2}) - \sqrt{1} + \sqrt{- 162}$

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$- 6 + i \cdot (6 \sqrt{2}) - \sqrt{1} + \sqrt{- 162}$

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

$- 6 + 6 \cdot i \sqrt{2} - \sqrt{1} + \sqrt{- 162}$

Any root of $1$ is $1$ .

$- 6 + 6 i \sqrt{2} - 1 \cdot 1 + \sqrt{- 162}$

Multiply $- 1$ by $1$ .

$- 6 + 6 i \sqrt{2} - 1 + \sqrt{- 162}$

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

$- 6 + 6 i \sqrt{2} - 1 + \sqrt{- 1 (162)}$

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

$- 6 + 6 i \sqrt{2} - 1 + \sqrt{- 1} \cdot \sqrt{162}$

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

$- 6 + 6 i \sqrt{2} - 1 + i \cdot \sqrt{162}$

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

Factor $81$ out of $162$ .

$- 6 + 6 i \sqrt{2} - 1 + i \cdot \sqrt{81 (2)}$

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

$- 6 + 6 i \sqrt{2} - 1 + i \cdot \sqrt{9^{2} \cdot 2}$

Pull terms out from under the radical.

$- 6 + 6 i \sqrt{2} - 1 + i \cdot (| 9 | \sqrt{2})$

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

$- 6 + 6 i \sqrt{2} - 1 + i \cdot (9 \sqrt{2})$

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

$- 6 + 6 i \sqrt{2} - 1 + 9 i \sqrt{2}$

Step 2

Subtract $1$ from $- 6$ .

$6 i \sqrt{2} - 7 + 9 i \sqrt{2}$

Step 3

Add $6 i \sqrt{2}$ and $9 i \sqrt{2}$ .

$15 i \sqrt{2} - 7$

Step 4

Reorder $15 i \sqrt{2}$ and $- 7$ .

$- 7 + 15 i \sqrt{2}$

Simplify the Radical Expression - square root of 36+ square root of -72- square root of 1+ square root of -162

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