Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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.

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.2

Simplify terms.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Cancel the common factor of $2$ .

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Step 1.2.2.1

Cancel the common factor.

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

Step 1.2.2.2

Rewrite the expression.

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

Step 1.2.3

Cancel the common factor of $2$ .

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Step 1.2.3.1

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

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

Step 1.2.3.2

Cancel the common factor.

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

Step 1.2.3.3

Rewrite the expression.

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

Step 1.2.4

Apply the distributive property.

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

Step 1.2.5

Multiply.

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Step 1.2.5.1

Multiply $4$ by $1$ .

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

Step 1.2.5.2

Multiply $4$ by $- 1$ .

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

Step 1.3

Simplify each term.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 2

Expand $(4 - 4 i \sqrt{3}) (\frac{\sqrt{3}}{2} + \frac{i}{2})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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

Cancel the common factor of $2$ .

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Step 3.1.1.1

Factor $2$ out of $4$ .

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

Step 3.1.1.2

Cancel the common factor.

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

Step 3.1.1.3

Rewrite the expression.

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

Step 3.1.2

Cancel the common factor of $2$ .

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Step 3.1.2.1

Factor $2$ out of $4$ .

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.1.3

Cancel the common factor of $2$ .

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Step 3.1.3.1

Factor $2$ out of $- 4 i \sqrt{3}$ .

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

Step 3.1.3.2

Cancel the common factor.

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

Step 3.1.3.3

Rewrite the expression.

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

Step 3.1.4

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

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

Step 3.1.5

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

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

Step 3.1.6

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

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

Step 3.1.7

Add $1$ and $1$ .

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

Step 3.1.8

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

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Step 3.1.8.1

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

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

Step 3.1.8.2

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

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

Step 3.1.8.3

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

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

Step 3.1.8.4

Cancel the common factor of $2$ .

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Step 3.1.8.4.1

Cancel the common factor.

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

Step 3.1.8.4.2

Rewrite the expression.

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

Step 3.1.8.5

Evaluate the exponent.

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

Step 3.1.9

Multiply $3$ by $- 2$ .

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

Step 3.1.10

Cancel the common factor of $2$ .

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Step 3.1.10.1

Factor $2$ out of $- 4 i \sqrt{3}$ .

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

Step 3.1.10.2

Cancel the common factor.

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

Step 3.1.10.3

Rewrite the expression.

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

Step 3.1.11

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

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

Step 3.1.12

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

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

Step 3.1.13

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

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

Step 3.1.14

Add $1$ and $1$ .

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

Step 3.1.15

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

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

Step 3.1.16

Multiply $- 2$ by $- 1$ .

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

Step 3.2

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

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

Step 3.3

Subtract $6 i$ from $2 i$ .

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

Step 4

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

$4 \sqrt{3} - 4 i$