Trigonometry Examples

$\sqrt{- 8^{2} \cdot 2^{2}}$

Step 1

Rewrite $8$ as $2^{3}$ .

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

Step 2

Multiply the exponents in ${(2^{3})}^{2}$ .

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Step 2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2

Multiply $3$ by $2$ .

$\sqrt{- (2^{2} \cdot 2^{6})}$

Step 3

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

$\sqrt{- 2^{2 + 6}}$

Step 4

Simplify the expression.

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Step 4.1

Add $2$ and $6$ .

$\sqrt{- 2^{8}}$

Step 4.2

Raise $2$ to the power of $8$ .

$\sqrt{- 1 \cdot 256}$

Step 4.3

Multiply $- 1$ by $256$ .

$\sqrt{- 256}$

Step 5

Rewrite $- 256$ as $- 1 (256)$ .

$\sqrt{- 1 (256)}$

Step 6

Rewrite $\sqrt{- 1 (256)}$ as $\sqrt{- 1} \cdot \sqrt{256}$ .

$\sqrt{- 1} \cdot \sqrt{256}$

Step 7

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

$i \cdot \sqrt{256}$

Step 8

Rewrite $256$ as $16^{2}$ .

$i \cdot \sqrt{16^{2}}$

Step 9

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

$i \cdot 16$

Step 10

Move $16$ to the left of $i$ .

$16 i$