Simplify 2(cos((5pi)/3)+isin((5pi)/3))*4(cos(pi/6)+isin(pi/6))

$2 (\cos (\frac{5 π}{3}) + i \sin (\frac{5 π}{3})) \cdot 4 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6}))$

Simplify terms.

Simplify each term.

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$2 (\cos (\frac{π}{3}) + i \sin (\frac{5 π}{3})) \cdot (4 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6})))$

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$2 (\frac{1}{2} + i \sin (\frac{5 π}{3})) \cdot (4 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6})))$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$2 (\frac{1}{2} + i (- \sin (\frac{π}{3}))) \cdot (4 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6})))$

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$2 (\frac{1}{2} + i (- \frac{\sqrt{3}}{2})) \cdot (4 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6})))$

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

$2 (\frac{1}{2} - \frac{i \sqrt{3}}{2}) \cdot (4 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6})))$

Simplify terms.

Apply the distributive property.

$(2 (\frac{1}{2}) + 2 (- \frac{i \sqrt{3}}{2})) \cdot (4 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6})))$

Cancel the common factor of $2$ .

Cancel the common factor.

$(2 (\frac{1}{2}) + 2 (- \frac{i \sqrt{3}}{2})) \cdot (4 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6})))$

Rewrite the expression.

$(1 + 2 (- \frac{i \sqrt{3}}{2})) \cdot (4 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6})))$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{i \sqrt{3}}{2}$ into the numerator.

$(1 + 2 \frac{- i \sqrt{3}}{2}) \cdot (4 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6})))$

Cancel the common factor.

$(1 + 2 \frac{- i \sqrt{3}}{2}) \cdot (4 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6})))$

Rewrite the expression.

$(1 - i \sqrt{3}) \cdot (4 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6})))$

Simplify each term.

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$(1 - i \sqrt{3}) \cdot (4 (\frac{\sqrt{3}}{2} + i \sin (\frac{π}{6})))$

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

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

Combine $i$ and $\frac{1}{2}$ .

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

Simplify terms.

Apply the distributive property.

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Expand $(1 - i \sqrt{3}) (2 \sqrt{3} + 2 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2 \sqrt{3}$ by $1$ .

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

Multiply $2 i$ by $1$ .

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

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

Multiply $2$ by $- 1$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

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

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $3$ by $- 2$ .

$2 \sqrt{3} + 2 i - 6 i - i \sqrt{3} (2 i)$

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

Multiply $2$ by $- 1$ .

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

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

$2 \sqrt{3} + 2 i - 6 i - 2 (i^{1} i) \sqrt{3}$

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

$2 \sqrt{3} + 2 i - 6 i - 2 (i^{1} i^{1}) \sqrt{3}$

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

$2 \sqrt{3} + 2 i - 6 i - 2 i^{1 + 1} \sqrt{3}$

Add $1$ and $1$ .

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

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

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

Multiply $- 2$ by $- 1$ .

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

Add $2 \sqrt{3}$ and $2 \sqrt{3}$ .

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

Subtract $6 i$ from $2 i$ .

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

Reorder $- 4 i$ and $4 \sqrt{3}$ .

$4 \sqrt{3} - 4 i$

Simplify 2(cos((5pi)/3)+isin((5pi)/3))*4(cos(pi/6)+isin(pi/6))

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