Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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 1.2

Simplify terms.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Cancel the common factor of $2$ .

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Step 1.2.2.1

Cancel the common factor.

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

Step 1.2.2.2

Rewrite the expression.

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

Step 1.2.3

Cancel the common factor of $2$ .

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Step 1.2.3.1

Cancel the common factor.

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

Step 1.2.3.2

Rewrite the expression.

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

Step 1.2.4

Apply the distributive property.

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

Step 1.2.5

Simplify the expression.

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Step 1.2.5.1

Multiply $5$ by $1$ .

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

Step 1.2.5.2

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

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

Step 1.3

Remove parentheses.

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

Step 1.4

Simplify each term.

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Step 1.4.1

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.

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

Step 1.4.2

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

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

Step 1.4.3

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

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

Step 1.4.4

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

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

Step 1.4.5

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

Multiply $- 1$ by $5$ .

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

Step 3.1.1.2

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

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

Step 3.1.2

Move the negative in front of the fraction.

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

Step 3.1.3

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

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

Step 3.1.4

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

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Step 3.1.4.1

Multiply $- 1$ by $5$ .

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

Step 3.1.4.2

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

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

Step 3.1.4.3

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

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

Step 3.1.4.4

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

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

Step 3.1.5

Move the negative in front of the fraction.

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

Step 3.1.6

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

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Step 3.1.6.1

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

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

Step 3.1.6.2

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

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

Step 3.1.6.3

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

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

Step 3.1.6.4

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

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

Step 3.1.6.5

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

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

Step 3.1.6.6

Add $1$ and $1$ .

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

Step 3.1.6.7

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

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

Step 3.1.6.8

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

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

Step 3.1.6.9

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

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

Step 3.1.6.10

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

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

Step 3.1.6.11

Add $1$ and $1$ .

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

Step 3.1.7

Simplify the numerator.

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Step 3.1.7.1

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

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

Step 3.1.7.2

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

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Step 3.1.7.2.1

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

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

Step 3.1.7.2.2

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

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

Step 3.1.7.2.3

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

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

Step 3.1.7.2.4

Cancel the common factor of $2$ .

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Step 3.1.7.2.4.1

Cancel the common factor.

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

Step 3.1.7.2.4.2

Rewrite the expression.

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

Step 3.1.7.2.5

Evaluate the exponent.

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

Step 3.1.7.3

Combine exponents.

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Step 3.1.7.3.1

Factor out negative.

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

Step 3.1.7.3.2

Multiply $5$ by $3$ .

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

Step 3.1.7.3.3

Multiply $- 1$ by $15$ .

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

Step 3.1.8

Move the negative in front of the fraction.

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 3.3

Subtract $15$ from $- 5$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Add $- 20$ and $0$ .

$\frac{- 20}{2}$

Step 4

Divide $- 20$ by $2$ .

$- 10$