Trigonometry Examples

$\frac{- 4 + \sqrt{- 8}}{\frac{- 4}{3}}$

Step 1

Pull out imaginary unit $i$ .

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Step 1.1

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

$\frac{- 4 + \sqrt{- 1 (8)}}{\frac{- 4}{3}}$

Step 1.2

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

$\frac{- 4 + \sqrt{- 1} \cdot \sqrt{8}}{\frac{- 4}{3}}$

Step 1.3

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

$\frac{- 4 + i \cdot \sqrt{8}}{\frac{- 4}{3}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$(- 4 + i \cdot \sqrt{8}) \frac{3}{- 4}$

Step 3

Simplify each term.

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Step 3.1

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

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Step 3.1.1

Factor $4$ out of $8$ .

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

Step 3.1.2

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

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

Step 3.2

Pull terms out from under the radical.

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

Step 3.3

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

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

Step 4

Simplify terms.

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Step 4.1

Move the negative in front of the fraction.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Cancel the common factor of $4$ .

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Step 4.3.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

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

Step 4.3.2

Factor $4$ out of $- 4$ .

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

Step 4.3.3

Cancel the common factor.

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

Step 4.3.4

Rewrite the expression.

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

Step 4.4

Multiply $- 1$ by $- 3$ .

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

Step 4.5

Cancel the common factor of $2$ .

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Step 4.5.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$3 + 2 i \sqrt{2} \frac{- 3}{4}$

Step 4.5.2

Factor $2$ out of $2 i \sqrt{2}$ .

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

Step 4.5.3

Factor $2$ out of $4$ .

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

Step 4.5.4

Cancel the common factor.

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

Step 4.5.5

Rewrite the expression.

$3 + i \sqrt{2} \frac{- 3}{2}$

Step 4.6

Combine $\frac{- 3}{2}$ and $i$ .

$3 + \frac{- 3 i}{2} \sqrt{2}$

Step 4.7

Combine $\frac{- 3 i}{2}$ and $\sqrt{2}$ .

$3 + \frac{- 3 i \sqrt{2}}{2}$

Step 4.8

Move the negative in front of the fraction.

$3 - \frac{3 i \sqrt{2}}{2}$

Step 5

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 \cdot \frac{2}{2} - \frac{3 i \sqrt{2}}{2}$

Step 6

Combine $3$ and $\frac{2}{2}$ .

$\frac{3 \cdot 2}{2} - \frac{3 i \sqrt{2}}{2}$

Step 7

Combine the numerators over the common denominator.

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