Trigonometry Examples

$\sqrt{- 12} (\sqrt{- 4} - \sqrt{2})$

Step 1

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

$\sqrt{- 1 (12)} (\sqrt{- 4} - \sqrt{2})$

Step 2

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

$\sqrt{- 1} \cdot \sqrt{12} (\sqrt{- 4} - \sqrt{2})$

Step 3

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

$i \cdot \sqrt{12} (\sqrt{- 4} - \sqrt{2})$

Step 4

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

$i \cdot \sqrt{12} (\sqrt{- 1 (4)} - \sqrt{2})$

Step 5

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

$i \cdot \sqrt{12} (\sqrt{- 1} \cdot \sqrt{4} - \sqrt{2})$

Step 6

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

$i \cdot \sqrt{12} (i \cdot \sqrt{4} - \sqrt{2})$

Step 7

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

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Step 7.1

Factor $4$ out of $12$ .

$i \cdot \sqrt{4 (3)} (i \cdot \sqrt{4} - \sqrt{2})$

Step 7.2

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

$i \cdot \sqrt{2^{2} \cdot 3} (i \cdot \sqrt{4} - \sqrt{2})$

Step 8

Simplify by cancelling exponent with radical.

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Step 8.1

Pull terms out from under the radical.

$i \cdot (2 \sqrt{3}) (i \cdot \sqrt{4} - \sqrt{2})$

Step 8.2

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

$2 \cdot i \sqrt{3} (i \cdot \sqrt{4} - \sqrt{2})$

Step 9

Simplify each term.

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Step 9.1

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

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

Step 9.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 9.3

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

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

Step 10

Apply the distributive property.

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

Step 11

Multiply $2 i \sqrt{3} (2 i)$ .

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Step 11.1

Multiply $2$ by $2$ .

$4 i \sqrt{3} i + 2 i \sqrt{3} (- \sqrt{2})$

Step 11.2

Raise $i$ to the power of $1$ .

$4 (i^{1} i) \sqrt{3} + 2 i \sqrt{3} (- \sqrt{2})$

Step 11.3

Raise $i$ to the power of $1$ .

$4 (i^{1} i^{1}) \sqrt{3} + 2 i \sqrt{3} (- \sqrt{2})$

Step 11.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 i^{1 + 1} \sqrt{3} + 2 i \sqrt{3} (- \sqrt{2})$

Step 11.5

Add $1$ and $1$ .

$4 i^{2} \sqrt{3} + 2 i \sqrt{3} (- \sqrt{2})$

Step 12

Multiply $2 i \sqrt{3} (- \sqrt{2})$ .

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Step 12.1

Multiply $- 1$ by $2$ .

$4 i^{2} \sqrt{3} - 2 i \sqrt{3} \sqrt{2}$

Step 12.2

Combine using the product rule for radicals.

$4 i^{2} \sqrt{3} - 2 i \sqrt{2 \cdot 3}$

Step 12.3

Multiply $2$ by $3$ .

$4 i^{2} \sqrt{3} - 2 i \sqrt{6}$

Step 13

Simplify each term.

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Step 13.1

Rewrite $i^{2}$ as $- 1$ .

$4 \cdot - 1 \sqrt{3} - 2 i \sqrt{6}$

Step 13.2

Multiply $4$ by $- 1$ .

$- 4 \sqrt{3} - 2 i \sqrt{6}$