Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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Step 1.7.1

Factor $4$ out of $8$ .

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

Step 1.7.2

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

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

Step 1.8

Pull terms out from under the radical.

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

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

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

Step 1.12

Pull terms out from under the radical.

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

Step 1.13

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

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

Step 1.14

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

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

Step 1.15

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

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

Step 1.16

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 $22$ .

$22 + i \sqrt{2}$