Algebra Examples

$\sqrt{25} - \sqrt{- 4} + \sqrt{9} - \sqrt{- 81}$

Step 1

Simplify each term.

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Step 1.1

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

$\sqrt{5^{2}} - \sqrt{- 4} + \sqrt{9} - \sqrt{- 81}$

Step 1.2

Pull terms out from under the radical.

$| 5 | - \sqrt{- 4} + \sqrt{9} - \sqrt{- 81}$

Step 1.3

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

$5 - \sqrt{- 4} + \sqrt{9} - \sqrt{- 81}$

Step 1.4

Rewrite $- 4$ as $- 1 (4)$ .

$5 - \sqrt{- 1 (4)} + \sqrt{9} - \sqrt{- 81}$

Step 1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$5 - (\sqrt{- 1} \cdot \sqrt{4}) + \sqrt{9} - \sqrt{- 81}$

Step 1.6

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

$5 - (i \cdot \sqrt{4}) + \sqrt{9} - \sqrt{- 81}$

Step 1.7

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

$5 - (i \cdot \sqrt{2^{2}}) + \sqrt{9} - \sqrt{- 81}$

Step 1.8

Pull terms out from under the radical.

$5 - (i \cdot | 2 |) + \sqrt{9} - \sqrt{- 81}$

Step 1.9

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

$5 - (i \cdot 2) + \sqrt{9} - \sqrt{- 81}$

Step 1.10

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

$5 - (2 \cdot i) + \sqrt{9} - \sqrt{- 81}$

Step 1.11

Multiply $2$ by $- 1$ .

$5 - 2 i + \sqrt{9} - \sqrt{- 81}$

Step 1.12

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

$5 - 2 i + \sqrt{3^{2}} - \sqrt{- 81}$

Step 1.13

Pull terms out from under the radical.

$5 - 2 i + | 3 | - \sqrt{- 81}$

Step 1.14

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

$5 - 2 i + 3 - \sqrt{- 81}$

Step 1.15

Rewrite $- 81$ as $- 1 (81)$ .

$5 - 2 i + 3 - \sqrt{- 1 (81)}$

Step 1.16

Rewrite $\sqrt{- 1 (81)}$ as $\sqrt{- 1} \cdot \sqrt{81}$ .

$5 - 2 i + 3 - (\sqrt{- 1} \cdot \sqrt{81})$

Step 1.17

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

$5 - 2 i + 3 - (i \cdot \sqrt{81})$

Step 1.18

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

$5 - 2 i + 3 - (i \cdot \sqrt{9^{2}})$

Step 1.19

Pull terms out from under the radical.

$5 - 2 i + 3 - (i \cdot | 9 |)$

Step 1.20

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

$5 - 2 i + 3 - (i \cdot 9)$

Step 1.21

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

$5 - 2 i + 3 - (9 \cdot i)$

Step 1.22

Multiply $9$ by $- 1$ .

$5 - 2 i + 3 - 9 i$

Step 2

Add $5$ and $3$ .

$8 - 2 i - 9 i$

Step 3

Subtract $9 i$ from $- 2 i$ .

$8 - 11 i$