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.

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

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.

Tap for more steps...

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 $2$ .

Tap for more steps...

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 $2$ .

Tap for more steps...

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 1.3

Simplify each term.

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.4

Simplify terms.

Tap for more steps...

Step 1.4.1

Apply the distributive property.

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

Step 1.4.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.4.2.1

Factor $2$ out of $4$ .

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

Step 1.4.2.2

Cancel the common factor.

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

Step 1.4.2.3

Rewrite the expression.

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

Step 1.4.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.4.3.1

Factor $2$ out of $4$ .

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

Step 1.4.3.2

Cancel the common factor.

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

Step 1.4.3.3

Rewrite the expression.

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

Step 2

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

Tap for more steps...

Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

Multiply $2 i$ by $1$ .

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

Step 3.1.3

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

Tap for more steps...

Step 3.1.3.1

Multiply $2$ by $- 1$ .

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

Step 3.1.3.2

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

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

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

Step 3.1.3.5

Add $1$ and $1$ .

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

Step 3.1.4

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

Tap for more steps...

Step 3.1.4.1

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

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

Step 3.1.4.2

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

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

Step 3.1.4.3

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

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

Step 3.1.4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.1.4.4.1

Cancel the common factor.

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

Step 3.1.4.4.2

Rewrite the expression.

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

Step 3.1.4.5

Evaluate the exponent.

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

Step 3.1.5

Multiply $3$ by $- 2$ .

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

Step 3.1.6

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

Tap for more steps...

Step 3.1.6.1

Multiply $2$ by $- 1$ .

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

Step 3.1.6.2

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

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

Step 3.1.6.3

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

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

Step 3.1.6.4

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 3.1.6.5

Add $1$ and $1$ .

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

Step 3.1.7

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

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

Step 3.1.8

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 $4 \sqrt{3}$ .

$4 \sqrt{3} - 4 i$