Algebra Examples

$\sqrt{81} + \sqrt{- 7} + \sqrt{9} + \sqrt{- 112}$

Step 1

Simplify each term.

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Step 1.1

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

$\sqrt{9^{2}} + \sqrt{- 7} + \sqrt{9} + \sqrt{- 112}$

Step 1.2

Pull terms out from under the radical.

$| 9 | + \sqrt{- 7} + \sqrt{9} + \sqrt{- 112}$

Step 1.3

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

$9 + \sqrt{- 7} + \sqrt{9} + \sqrt{- 112}$

Step 1.4

Rewrite $- 7$ as $- 1 (7)$ .

$9 + \sqrt{- 1 (7)} + \sqrt{9} + \sqrt{- 112}$

Step 1.5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$9 + \sqrt{- 1} \cdot \sqrt{7} + \sqrt{9} + \sqrt{- 112}$

Step 1.6

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

$9 + i \cdot \sqrt{7} + \sqrt{9} + \sqrt{- 112}$

Step 1.7

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

$9 + i \sqrt{7} + \sqrt{3^{2}} + \sqrt{- 112}$

Step 1.8

Pull terms out from under the radical.

$9 + i \sqrt{7} + | 3 | + \sqrt{- 112}$

Step 1.9

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

$9 + i \sqrt{7} + 3 + \sqrt{- 112}$

Step 1.10

Rewrite $- 112$ as $- 1 (112)$ .

$9 + i \sqrt{7} + 3 + \sqrt{- 1 (112)}$

Step 1.11

Rewrite $\sqrt{- 1 (112)}$ as $\sqrt{- 1} \cdot \sqrt{112}$ .

$9 + i \sqrt{7} + 3 + \sqrt{- 1} \cdot \sqrt{112}$

Step 1.12

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

$9 + i \sqrt{7} + 3 + i \cdot \sqrt{112}$

Step 1.13

Rewrite $112$ as $4^{2} \cdot 7$ .

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Step 1.13.1

Factor $16$ out of $112$ .

$9 + i \sqrt{7} + 3 + i \cdot \sqrt{16 (7)}$

Step 1.13.2

Rewrite $16$ as $4^{2}$ .

$9 + i \sqrt{7} + 3 + i \cdot \sqrt{4^{2} \cdot 7}$

Step 1.14

Pull terms out from under the radical.

$9 + i \sqrt{7} + 3 + i \cdot (| 4 | \sqrt{7})$

Step 1.15

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$9 + i \sqrt{7} + 3 + i \cdot (4 \sqrt{7})$

Step 1.16

Move $4$ to the left of $i$ .

$9 + i \sqrt{7} + 3 + 4 i \sqrt{7}$

Step 2

Add $9$ and $3$ .

$i \sqrt{7} + 12 + 4 i \sqrt{7}$

Step 3

Add $i \sqrt{7}$ and $4 i \sqrt{7}$ .

$5 i \sqrt{7} + 12$

Step 4

Reorder $5 i \sqrt{7}$ and $12$ .

$12 + 5 i \sqrt{7}$