Precalculus Examples

$\sqrt{- 12} - (\sqrt{- 11} - \sqrt{5})$

Step 1

Simplify each term.

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Step 1.1

Rewrite $- 12$ as $- 1 (12)$ .

$\sqrt{- 1 (12)} - (\sqrt{- 11} - \sqrt{5})$

Step 1.2

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$\sqrt{- 1} \cdot \sqrt{12} - (\sqrt{- 11} - \sqrt{5})$

Step 1.3

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

$i \cdot \sqrt{12} - (\sqrt{- 11} - \sqrt{5})$

Step 1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.4.1

Factor $4$ out of $12$ .

$i \cdot \sqrt{4 (3)} - (\sqrt{- 11} - \sqrt{5})$

Step 1.4.2

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

$i \cdot \sqrt{2^{2} \cdot 3} - (\sqrt{- 11} - \sqrt{5})$

Step 1.5

Pull terms out from under the radical.

$i \cdot (2 \sqrt{3}) - (\sqrt{- 11} - \sqrt{5})$

Step 1.6

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

$2 \cdot i \sqrt{3} - (\sqrt{- 11} - \sqrt{5})$

Step 1.7

Rewrite $- 11$ as $- 1 (11)$ .

$2 i \sqrt{3} - (\sqrt{- 1 (11)} - \sqrt{5})$

Step 1.8

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$2 i \sqrt{3} - (\sqrt{- 1} \cdot \sqrt{11} - \sqrt{5})$

Step 1.9

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

$2 i \sqrt{3} - (i \cdot \sqrt{11} - \sqrt{5})$

Step 1.10

Apply the distributive property.

$2 i \sqrt{3} - (i \sqrt{11}) - - \sqrt{5}$

Step 1.11

Multiply $- - \sqrt{5}$ .

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Step 1.11.1

Multiply $- 1$ by $- 1$ .

$2 i \sqrt{3} - i \sqrt{11} + 1 \sqrt{5}$

Step 1.11.2

Multiply $\sqrt{5}$ by $1$ .

$2 i \sqrt{3} - i \sqrt{11} + \sqrt{5}$

Step 2

Reorder $2 i \sqrt{3} - i \sqrt{11}$ and $\sqrt{5}$ .

$\sqrt{5} + 2 i \sqrt{3} - i \sqrt{11}$