Algebra Examples

$- \sqrt{81} + \sqrt{- 28} + \sqrt{1} - \sqrt{- 175}$

Step 1

Simplify each term.

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Step 1.1

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

$- \sqrt{9^{2}} + \sqrt{- 28} + \sqrt{1} - \sqrt{- 175}$

Step 1.2

Pull terms out from under the radical.

$- | 9 | + \sqrt{- 28} + \sqrt{1} - \sqrt{- 175}$

Step 1.3

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

$- 1 \cdot 9 + \sqrt{- 28} + \sqrt{1} - \sqrt{- 175}$

Step 1.4

Multiply $- 1$ by $9$ .

$- 9 + \sqrt{- 28} + \sqrt{1} - \sqrt{- 175}$

Step 1.5

Rewrite $- 28$ as $- 1 (28)$ .

$- 9 + \sqrt{- 1 (28)} + \sqrt{1} - \sqrt{- 175}$

Step 1.6

Rewrite $\sqrt{- 1 (28)}$ as $\sqrt{- 1} \cdot \sqrt{28}$ .

$- 9 + \sqrt{- 1} \cdot \sqrt{28} + \sqrt{1} - \sqrt{- 175}$

Step 1.7

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

$- 9 + i \cdot \sqrt{28} + \sqrt{1} - \sqrt{- 175}$

Step 1.8

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 1.8.1

Factor $4$ out of $28$ .

$- 9 + i \cdot \sqrt{4 (7)} + \sqrt{1} - \sqrt{- 175}$

Step 1.8.2

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

$- 9 + i \cdot \sqrt{2^{2} \cdot 7} + \sqrt{1} - \sqrt{- 175}$

Step 1.9

Pull terms out from under the radical.

$- 9 + i \cdot (| 2 | \sqrt{7}) + \sqrt{1} - \sqrt{- 175}$

Step 1.10

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

$- 9 + i \cdot (2 \sqrt{7}) + \sqrt{1} - \sqrt{- 175}$

Step 1.11

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

$- 9 + 2 \cdot i \sqrt{7} + \sqrt{1} - \sqrt{- 175}$

Step 1.12

Any root of $1$ is $1$ .

$- 9 + 2 i \sqrt{7} + 1 - \sqrt{- 175}$

Step 1.13

Rewrite $- 175$ as $- 1 (175)$ .

$- 9 + 2 i \sqrt{7} + 1 - \sqrt{- 1 (175)}$

Step 1.14

Rewrite $\sqrt{- 1 (175)}$ as $\sqrt{- 1} \cdot \sqrt{175}$ .

$- 9 + 2 i \sqrt{7} + 1 - (\sqrt{- 1} \cdot \sqrt{175})$

Step 1.15

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

$- 9 + 2 i \sqrt{7} + 1 - (i \cdot \sqrt{175})$

Step 1.16

Rewrite $175$ as $5^{2} \cdot 7$ .

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Step 1.16.1

Factor $25$ out of $175$ .

$- 9 + 2 i \sqrt{7} + 1 - (i \cdot \sqrt{25 (7)})$

Step 1.16.2

Rewrite $25$ as $5^{2}$ .

$- 9 + 2 i \sqrt{7} + 1 - (i \cdot \sqrt{5^{2} \cdot 7})$

Step 1.17

Pull terms out from under the radical.

$- 9 + 2 i \sqrt{7} + 1 - (i \cdot (| 5 | \sqrt{7}))$

Step 1.18

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$- 9 + 2 i \sqrt{7} + 1 - (i \cdot (5 \sqrt{7}))$

Step 1.19

Move $5$ to the left of $i$ .

$- 9 + 2 i \sqrt{7} + 1 - (5 \cdot i \sqrt{7})$

Step 1.20

Multiply $5$ by $- 1$ .

$- 9 + 2 i \sqrt{7} + 1 - 5 i \sqrt{7}$

Step 2

Add $- 9$ and $1$ .

$2 i \sqrt{7} - 8 - 5 i \sqrt{7}$

Step 3

Subtract $5 i \sqrt{7}$ from $2 i \sqrt{7}$ .

$- 3 i \sqrt{7} - 8$

Step 4

Reorder $- 3 i \sqrt{7}$ and $- 8$ .

$- 8 - 3 i \sqrt{7}$