Trigonometry Examples

$\sqrt{- 8} \sqrt{- 4}$

Step 1

Rewrite $- 8$ as $- 1 (8)$ .

$\sqrt{- 1 (8)} \sqrt{- 4}$

Step 2

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$\sqrt{- 1} \cdot \sqrt{8} \sqrt{- 4}$

Step 3

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

$i \cdot \sqrt{8} \sqrt{- 4}$

Step 4

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

$i \cdot \sqrt{8} \sqrt{- 1 (4)}$

Step 5

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

$i \cdot \sqrt{8} (\sqrt{- 1} \cdot \sqrt{4})$

Step 6

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

$i \cdot \sqrt{8} (i \cdot \sqrt{4})$

Step 7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 7.1

Factor $4$ out of $8$ .

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

Step 7.2

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

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

Step 8

Pull terms out from under the radical.

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

Step 9

Simplify the expression.

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Step 9.1

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

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

Step 9.2

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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Step 12.1

Multiply $2$ by $2$ .

$4 i \sqrt{2} i$

Step 12.2

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

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

Step 12.3

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

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

Step 12.4

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

$4 i^{1 + 1} \sqrt{2}$

Step 12.5

Add $1$ and $1$ .

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

Step 13

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

$4 \cdot - 1 \sqrt{2}$

Step 14

Multiply $4$ by $- 1$ .

$- 4 \sqrt{2}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$- 4 \sqrt{2}$

Decimal Form:

$- 5.65685424 \dots$