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

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

Step 1

Simplify each term.

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

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

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

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

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

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

Step 2

Simplify terms.

Apply the distributive property.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

Simplify each term.

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

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

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

$(1 + i \sqrt{3}) \cdot (5 (- \frac{1}{2} + i \sin (\frac{2 π}{3})))$

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

$(1 + i \sqrt{3}) \cdot (5 (- \frac{1}{2} + i \sin (\frac{π}{3})))$

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

$(1 + i \sqrt{3}) \cdot (5 (- \frac{1}{2} + i \frac{\sqrt{3}}{2}))$

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

$(1 + i \sqrt{3}) \cdot (5 (- \frac{1}{2} + \frac{i \sqrt{3}}{2}))$

Step 4

Apply the distributive property.

$(1 + i \sqrt{3}) \cdot (5 (- \frac{1}{2}) + 5 \frac{i \sqrt{3}}{2})$

Step 5

Multiply $5 (- \frac{1}{2})$ .

Multiply $- 1$ by $5$ .

$(1 + i \sqrt{3}) \cdot (- 5 (\frac{1}{2}) + 5 \frac{i \sqrt{3}}{2})$

Combine $- 5$ and $\frac{1}{2}$ .

$(1 + i \sqrt{3}) \cdot (\frac{- 5}{2} + 5 \frac{i \sqrt{3}}{2})$

Step 6

Simplify terms.

Combine $5$ and $\frac{i \sqrt{3}}{2}$ .

$(1 + i \sqrt{3}) \cdot (\frac{- 5}{2} + \frac{5 (i \sqrt{3})}{2})$

Move the negative in front of the fraction.

$(1 + i \sqrt{3}) \cdot (- \frac{5}{2} + \frac{5 i \sqrt{3}}{2})$

Step 7

Expand $(1 + i \sqrt{3}) (- \frac{5}{2} + \frac{5 i \sqrt{3}}{2})$ using the FOIL Method.

Apply the distributive property.

$1 (- \frac{5}{2} + \frac{5 i \sqrt{3}}{2}) + i \sqrt{3} (- \frac{5}{2} + \frac{5 i \sqrt{3}}{2})$

Apply the distributive property.

$1 (- \frac{5}{2}) + 1 \frac{5 i \sqrt{3}}{2} + i \sqrt{3} (- \frac{5}{2} + \frac{5 i \sqrt{3}}{2})$

Apply the distributive property.

$1 (- \frac{5}{2}) + 1 \frac{5 i \sqrt{3}}{2} + i \sqrt{3} (- \frac{5}{2}) + i \sqrt{3} \frac{5 i \sqrt{3}}{2}$

Step 8

Simplify and combine like terms.

Simplify each term.

Multiply $- \frac{5}{2}$ by $1$ .

$- \frac{5}{2} + 1 \frac{5 i \sqrt{3}}{2} + i \sqrt{3} (- \frac{5}{2}) + i \sqrt{3} \frac{5 i \sqrt{3}}{2}$

Multiply $\frac{5 i \sqrt{3}}{2}$ by $1$ .

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} + i \sqrt{3} (- \frac{5}{2}) + i \sqrt{3} \frac{5 i \sqrt{3}}{2}$

Multiply $i \sqrt{3} (- \frac{5}{2})$ .

Combine $i$ and $\frac{5}{2}$ .

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} + \sqrt{3} (- \frac{i \cdot 5}{2}) + i \sqrt{3} \frac{5 i \sqrt{3}}{2}$

Combine $\sqrt{3}$ and $\frac{i \cdot 5}{2}$ .

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{\sqrt{3} (i \cdot 5)}{2} + i \sqrt{3} \frac{5 i \sqrt{3}}{2}$

Move $5$ to the left of $\sqrt{3} i$ .

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + i \sqrt{3} \frac{5 i \sqrt{3}}{2}$

Multiply $i \sqrt{3} \frac{5 i \sqrt{3}}{2}$ .

Combine $\frac{5 i \sqrt{3}}{2}$ and $i$ .

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{5 i \sqrt{3} i}{2} \sqrt{3}$

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

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{5 (i^{1} i) \sqrt{3}}{2} \sqrt{3}$

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

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{5 (i^{1} i^{1}) \sqrt{3}}{2} \sqrt{3}$

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

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{5 i^{1 + 1} \sqrt{3}}{2} \sqrt{3}$

Add $1$ and $1$ .

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{5 i^{2} \sqrt{3}}{2} \sqrt{3}$

Combine $\frac{5 i^{2} \sqrt{3}}{2}$ and $\sqrt{3}$ .

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{5 i^{2} \sqrt{3} \sqrt{3}}{2}$

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

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{5 i^{2} ({\sqrt{3}}^{1} \sqrt{3})}{2}$

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

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{5 i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{2}$

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

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{5 i^{2} {\sqrt{3}}^{1 + 1}}{2}$

Add $1$ and $1$ .

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{5 i^{2} {\sqrt{3}}^{2}}{2}$

Simplify the numerator.

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

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{5 \cdot - 1 {\sqrt{3}}^{2}}{2}$

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

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

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{5 \cdot - 1 {(3^{\frac{1}{2}})}^{2}}{2}$

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

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{5 \cdot - 1 \cdot 3^{\frac{1}{2} \cdot 2}}{2}$

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

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{5 \cdot - 1 \cdot 3^{\frac{2}{2}}}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{5 \cdot - 1 \cdot 3^{\frac{2}{2}}}{2}$

Rewrite the expression.

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{5 \cdot - 1 \cdot 3^{1}}{2}$

Evaluate the exponent.

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{5 \cdot - 1 \cdot 3}{2}$

Combine exponents.

Factor out negative.

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{- (5 \cdot 3)}{2}$

Multiply $5$ by $3$ .

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{- 1 \cdot 15}{2}$

Multiply $- 1$ by $15$ .

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} + \frac{- 15}{2}$

Move the negative in front of the fraction.

$- \frac{5}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2} - \frac{15}{2}$

Combine the numerators over the common denominator.

$\frac{- 5 - 15}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2}$

Subtract $15$ from $- 5$ .

$\frac{- 20}{2} + \frac{5 i \sqrt{3}}{2} - \frac{5 \sqrt{3} i}{2}$

Step 9

Combine the numerators over the common denominator.

$\frac{- 20 + 5 i \sqrt{3} - 5 \sqrt{3} i}{2}$

Step 10

Reorder the factors of $- 5 \sqrt{3} i$ .

$\frac{- 20 + 5 i \sqrt{3} - 5 i \sqrt{3}}{2}$

Step 11

Subtract $5 i \sqrt{3}$ from $5 i \sqrt{3}$ .

$\frac{- 20 + 0}{2}$

Step 12

Simplify the expression.

Add $- 20$ and $0$ .

$\frac{- 20}{2}$

Divide $- 20$ by $2$ .

$- 10$

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

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