Algebra Examples

$- \sqrt{4} + \sqrt{- 75} - \sqrt{9} + \sqrt{- 27}$

Step 1

Simplify each term.

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Step 1.1

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

$- \sqrt{2^{2}} + \sqrt{- 75} - \sqrt{9} + \sqrt{- 27}$

Step 1.2

Pull terms out from under the radical.

$- | 2 | + \sqrt{- 75} - \sqrt{9} + \sqrt{- 27}$

Step 1.3

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

$- 1 \cdot 2 + \sqrt{- 75} - \sqrt{9} + \sqrt{- 27}$

Step 1.4

Multiply $- 1$ by $2$ .

$- 2 + \sqrt{- 75} - \sqrt{9} + \sqrt{- 27}$

Step 1.5

Rewrite $- 75$ as $- 1 (75)$ .

$- 2 + \sqrt{- 1 (75)} - \sqrt{9} + \sqrt{- 27}$

Step 1.6

Rewrite $\sqrt{- 1 (75)}$ as $\sqrt{- 1} \cdot \sqrt{75}$ .

$- 2 + \sqrt{- 1} \cdot \sqrt{75} - \sqrt{9} + \sqrt{- 27}$

Step 1.7

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

$- 2 + i \cdot \sqrt{75} - \sqrt{9} + \sqrt{- 27}$

Step 1.8

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 1.8.1

Factor $25$ out of $75$ .

$- 2 + i \cdot \sqrt{25 (3)} - \sqrt{9} + \sqrt{- 27}$

Step 1.8.2

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

$- 2 + i \cdot \sqrt{5^{2} \cdot 3} - \sqrt{9} + \sqrt{- 27}$

Step 1.9

Pull terms out from under the radical.

$- 2 + i \cdot (| 5 | \sqrt{3}) - \sqrt{9} + \sqrt{- 27}$

Step 1.10

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

$- 2 + i \cdot (5 \sqrt{3}) - \sqrt{9} + \sqrt{- 27}$

Step 1.11

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

$- 2 + 5 \cdot i \sqrt{3} - \sqrt{9} + \sqrt{- 27}$

Step 1.12

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

$- 2 + 5 i \sqrt{3} - \sqrt{3^{2}} + \sqrt{- 27}$

Step 1.13

Pull terms out from under the radical.

$- 2 + 5 i \sqrt{3} - | 3 | + \sqrt{- 27}$

Step 1.14

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

$- 2 + 5 i \sqrt{3} - 1 \cdot 3 + \sqrt{- 27}$

Step 1.15

Multiply $- 1$ by $3$ .

$- 2 + 5 i \sqrt{3} - 3 + \sqrt{- 27}$

Step 1.16

Rewrite $- 27$ as $- 1 (27)$ .

$- 2 + 5 i \sqrt{3} - 3 + \sqrt{- 1 (27)}$

Step 1.17

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$- 2 + 5 i \sqrt{3} - 3 + \sqrt{- 1} \cdot \sqrt{27}$

Step 1.18

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

$- 2 + 5 i \sqrt{3} - 3 + i \cdot \sqrt{27}$

Step 1.19

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.19.1

Factor $9$ out of $27$ .

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

Step 1.19.2

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

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

Step 1.20

Pull terms out from under the radical.

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

Step 1.21

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

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

Step 1.22

Move $3$ to the left of $i$ .

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

Step 2

Subtract $3$ from $- 2$ .

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

Step 3

Add $5 i \sqrt{3}$ and $3 i \sqrt{3}$ .

$8 i \sqrt{3} - 5$

Step 4

Reorder $8 i \sqrt{3}$ and $- 5$ .

$- 5 + 8 i \sqrt{3}$