Algebra Examples

$\sqrt{9} + \sqrt{- 27} + \sqrt{100} - \sqrt{- 3}$

Step 1

Simplify each term.

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Step 1.1

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

$\sqrt{3^{2}} + \sqrt{- 27} + \sqrt{100} - \sqrt{- 3}$

Step 1.2

Pull terms out from under the radical.

$| 3 | + \sqrt{- 27} + \sqrt{100} - \sqrt{- 3}$

Step 1.3

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

$3 + \sqrt{- 27} + \sqrt{100} - \sqrt{- 3}$

Step 1.4

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

$3 + \sqrt{- 1 (27)} + \sqrt{100} - \sqrt{- 3}$

Step 1.5

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

$3 + \sqrt{- 1} \cdot \sqrt{27} + \sqrt{100} - \sqrt{- 3}$

Step 1.6

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

$3 + i \cdot \sqrt{27} + \sqrt{100} - \sqrt{- 3}$

Step 1.7

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

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Step 1.7.1

Factor $9$ out of $27$ .

$3 + i \cdot \sqrt{9 (3)} + \sqrt{100} - \sqrt{- 3}$

Step 1.7.2

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

$3 + i \cdot \sqrt{3^{2} \cdot 3} + \sqrt{100} - \sqrt{- 3}$

Step 1.8

Pull terms out from under the radical.

$3 + i \cdot (| 3 | \sqrt{3}) + \sqrt{100} - \sqrt{- 3}$

Step 1.9

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

$3 + i \cdot (3 \sqrt{3}) + \sqrt{100} - \sqrt{- 3}$

Step 1.10

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

$3 + 3 \cdot i \sqrt{3} + \sqrt{100} - \sqrt{- 3}$

Step 1.11

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

$3 + 3 i \sqrt{3} + \sqrt{10^{2}} - \sqrt{- 3}$

Step 1.12

Pull terms out from under the radical.

$3 + 3 i \sqrt{3} + | 10 | - \sqrt{- 3}$

Step 1.13

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

$3 + 3 i \sqrt{3} + 10 - \sqrt{- 3}$

Step 1.14

Rewrite $- 3$ as $- 1 (3)$ .

$3 + 3 i \sqrt{3} + 10 - \sqrt{- 1 (3)}$

Step 1.15

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$3 + 3 i \sqrt{3} + 10 - (\sqrt{- 1} \cdot \sqrt{3})$

Step 1.16

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

$3 + 3 i \sqrt{3} + 10 - i \sqrt{3}$

Step 2

Add $3$ and $10$ .

$3 i \sqrt{3} + 13 - i \sqrt{3}$

Step 3

Subtract $i \sqrt{3}$ from $3 i \sqrt{3}$ .

$2 i \sqrt{3} + 13$

Step 4

Reorder $2 i \sqrt{3}$ and $13$ .

$13 + 2 i \sqrt{3}$