Evaluate (6(cos(pi/3)+isin(pi/3)))/(3(cos(pi/6)+isin(pi/6)))

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

Step 1

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6 (\cos (\frac{π}{3}) + i \sin (\frac{π}{3}))$ .

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

Cancel the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

Step 2

Multiply the numerator and denominator of $\frac{2 (\cos (\frac{π}{3}) + i \sin (\frac{π}{3}))}{0.8660254 + 0.5 i}$ by the conjugate of $0.8660254 + 0.5 i$ to make the denominator real.

$\frac{2 (\cos (\frac{π}{3}) + i \sin (\frac{π}{3}))}{0.8660254 + 0.5 i} \cdot \frac{0.8660254 - 0.5 i}{0.8660254 - 0.5 i}$

Step 3

Multiply.

Combine.

$\frac{2 (\cos (\frac{π}{3}) + i \sin (\frac{π}{3})) (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Simplify the numerator.

Simplify each term.

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

$\frac{2 (\frac{1}{2} + i \sin (\frac{π}{3})) (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

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

$\frac{2 (\frac{1}{2} + i \frac{\sqrt{3}}{2}) (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

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

$\frac{2 (\frac{1}{2} + \frac{i \sqrt{3}}{2}) (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Apply the distributive property.

$\frac{(2 (\frac{1}{2}) + 2 \frac{i \sqrt{3}}{2}) (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{(2 (\frac{1}{2}) + 2 \frac{i \sqrt{3}}{2}) (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Rewrite the expression.

$\frac{(1 + 2 \frac{i \sqrt{3}}{2}) (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{(1 + 2 \frac{i \sqrt{3}}{2}) (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Rewrite the expression.

$\frac{(1 + i \sqrt{3}) (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Expand $(1 + i \sqrt{3}) (0.8660254 - 0.5 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{1 (0.8660254 - 0.5 i) + i \sqrt{3} (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Apply the distributive property.

$\frac{1 \cdot 0.8660254 + 1 (- 0.5 i) + i \sqrt{3} (0.8660254 - 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Apply the distributive property.

$\frac{1 \cdot 0.8660254 + 1 (- 0.5 i) + i \sqrt{3} \cdot 0.8660254 + i \sqrt{3} (- 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $0.8660254$ by $1$ .

$\frac{0.8660254 + 1 (- 0.5 i) + i \sqrt{3} \cdot 0.8660254 + i \sqrt{3} (- 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Multiply $- 0.5 i$ by $1$ .

$\frac{0.8660254 - 0.5 i + i \sqrt{3} \cdot 0.8660254 + i \sqrt{3} (- 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Multiply $0.8660254$ by $\sqrt{3}$ .

$\frac{0.8660254 - 0.5 i + i \cdot 1.5 + i \sqrt{3} (- 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Move $1.5$ to the left of $i$ .

$\frac{0.8660254 - 0.5 i + 1.5 \cdot i + i \sqrt{3} (- 0.5 i)}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Multiply $i \sqrt{3} (- 0.5 i)$ .

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

$\frac{0.8660254 - 0.5 i + 1.5 i + \sqrt{3} (- 0.5 (i^{1} i))}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

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

$\frac{0.8660254 - 0.5 i + 1.5 i + \sqrt{3} (- 0.5 (i^{1} i^{1}))}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

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

$\frac{0.8660254 - 0.5 i + 1.5 i + \sqrt{3} (- 0.5 i^{1 + 1})}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Add $1$ and $1$ .

$\frac{0.8660254 - 0.5 i + 1.5 i + \sqrt{3} (- 0.5 i^{2})}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Multiply $- 0.5$ by $\sqrt{3}$ .

$\frac{0.8660254 - 0.5 i + 1.5 i - 0.8660254 i^{2}}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

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

$\frac{0.8660254 - 0.5 i + 1.5 i - 0.8660254 \cdot - 1}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Multiply $- 0.8660254$ by $- 1$ .

$\frac{0.8660254 - 0.5 i + 1.5 i + 0.8660254}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Add $0.8660254$ and $0.8660254$ .

$\frac{1.7320508 - 0.5 i + 1.5 i}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Add $- 0.5 i$ and $1.5 i$ .

$\frac{1.7320508 + 1 i}{(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)}$

Simplify the denominator.

Expand $(0.8660254 + 0.5 i) (0.8660254 - 0.5 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{1.7320508 + 1 i}{0.8660254 (0.8660254 - 0.5 i) + 0.5 i (0.8660254 - 0.5 i)}$

Apply the distributive property.

$\frac{1.7320508 + 1 i}{0.8660254 \cdot 0.8660254 + 0.8660254 (- 0.5 i) + 0.5 i (0.8660254 - 0.5 i)}$

Apply the distributive property.

$\frac{1.7320508 + 1 i}{0.8660254 \cdot 0.8660254 + 0.8660254 (- 0.5 i) + 0.5 i \cdot 0.8660254 + 0.5 i (- 0.5 i)}$

Simplify.

Multiply $0.8660254$ by $0.8660254$ .

$\frac{1.7320508 + 1 i}{0.75 + 0.8660254 (- 0.5 i) + 0.5 i \cdot 0.8660254 + 0.5 i (- 0.5 i)}$

Multiply $- 0.5$ by $0.8660254$ .

$\frac{1.7320508 + 1 i}{0.75 - 0.4330127 i + 0.5 i \cdot 0.8660254 + 0.5 i (- 0.5 i)}$

Multiply $0.8660254$ by $0.5$ .

$\frac{1.7320508 + 1 i}{0.75 - 0.4330127 i + 0.4330127 i + 0.5 i (- 0.5 i)}$

Multiply $- 0.5$ by $0.5$ .

$\frac{1.7320508 + 1 i}{0.75 - 0.4330127 i + 0.4330127 i - 0.25 i i}$

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

$\frac{1.7320508 + 1 i}{0.75 - 0.4330127 i + 0.4330127 i - 0.25 (i^{1} i)}$

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

$\frac{1.7320508 + 1 i}{0.75 - 0.4330127 i + 0.4330127 i - 0.25 (i^{1} i^{1})}$

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

$\frac{1.7320508 + 1 i}{0.75 - 0.4330127 i + 0.4330127 i - 0.25 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{1.7320508 + 1 i}{0.75 - 0.4330127 i + 0.4330127 i - 0.25 i^{2}}$

Add $- 0.4330127 i$ and $0.4330127 i$ .

$\frac{1.7320508 + 1 i}{0.75 + 0 i - 0.25 i^{2}}$

Simplify each term.

Multiply $0$ by $i$ .

$\frac{1.7320508 + 1 i}{0.75 + 0 - 0.25 i^{2}}$

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

$\frac{1.7320508 + 1 i}{0.75 + 0 - 0.25 \cdot - 1}$

Multiply $- 0.25$ by $- 1$ .

$\frac{1.7320508 + 1 i}{0.75 + 0 + 0.25}$

Add $0.75$ and $0$ .

$\frac{1.7320508 + 1 i}{0.75 + 0.25}$

Add $0.75$ and $0.25$ .

$\frac{1.7320508 + 1 i}{1}$

Step 4

Rewrite $1.7320508$ as $1 (1.7320508)$ .

$\frac{1 (1.7320508) + 1 i}{1}$

Step 5

Factor $1$ out of $1 i$ .

$\frac{1 (1.7320508) + 1 (1 i)}{1}$

Step 6

Factor $1$ out of $1 (1.7320508) + 1 (1 i)$ .

$\frac{1 (1.7320508 + 1 i)}{1}$

Step 7

Factor $1$ out of $1$ .

$\frac{1 (1.7320508 + 1 i)}{1 (1)}$

Step 8

Separate fractions.

$\frac{1}{1} \cdot \frac{1.7320508 + 1 i}{1}$

Step 9

Simplify the expression.

Divide $1$ by $1$ .

$1 \frac{1.7320508 + 1 i}{1}$

Divide $1.7320508 + 1 i$ by $1$ .

$1 (1.7320508 + 1 i)$

Step 10

Apply the distributive property.

$1 \cdot 1.7320508 + 1 (1 i)$

Step 11

Multiply.

Multiply $1$ by $1.7320508$ .

$1.7320508 + 1 (1 i)$

Multiply $1$ by $1$ .

$1.7320508 + 1 i$

Evaluate (6(cos(pi/3)+isin(pi/3)))/(3(cos(pi/6)+isin(pi/6)))

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