Basic Math Examples

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

Step 1

Multiply the numerator and denominator of $\frac{1 + i \sqrt{3}}{2 - 6 i}$ by the conjugate of $2 - 6 i$ to make the denominator real.

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

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand $(1 + i \sqrt{3}) (2 + 6 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Multiply $2$ by $1$ .

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

Step 2.2.2.2

Multiply $6 i$ by $1$ .

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

Step 2.2.2.3

Move $2$ to the left of $i \sqrt{3}$ .

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

Step 2.2.2.4

Multiply $i \sqrt{3} (6 i)$ .

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Step 2.2.2.4.1

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

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

Step 2.2.2.4.2

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

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

Step 2.2.2.4.3

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

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

Step 2.2.2.4.4

Add $1$ and $1$ .

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

Step 2.2.2.5

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

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

Step 2.2.2.6

Multiply $6$ by $- 1$ .

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

Step 2.2.2.7

Move $- 6$ to the left of $\sqrt{3}$ .

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

Step 2.2.3

Reorder $2 + 6 i + 2 i \sqrt{3}$ and $- 6 \sqrt{3}$ .

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

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(2 - 6 i) (2 + 6 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

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

Step 2.3.1.2

Apply the distributive property.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $2$ by $2$ .

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

Step 2.3.2.2

Multiply $6$ by $2$ .

$\frac{- 6 \sqrt{3} + 2 + 6 i + 2 i \sqrt{3}}{4 + 12 i - 6 i \cdot 2 - 6 i (6 i)}$

Step 2.3.2.3

Multiply $2$ by $- 6$ .

$\frac{- 6 \sqrt{3} + 2 + 6 i + 2 i \sqrt{3}}{4 + 12 i - 12 i - 6 i (6 i)}$

Step 2.3.2.4

Multiply $6$ by $- 6$ .

$\frac{- 6 \sqrt{3} + 2 + 6 i + 2 i \sqrt{3}}{4 + 12 i - 12 i - 36 i i}$

Step 2.3.2.5

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

$\frac{- 6 \sqrt{3} + 2 + 6 i + 2 i \sqrt{3}}{4 + 12 i - 12 i - 36 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{- 6 \sqrt{3} + 2 + 6 i + 2 i \sqrt{3}}{4 + 12 i - 12 i - 36 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{- 6 \sqrt{3} + 2 + 6 i + 2 i \sqrt{3}}{4 + 12 i - 12 i - 36 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{- 6 \sqrt{3} + 2 + 6 i + 2 i \sqrt{3}}{4 + 12 i - 12 i - 36 i^{2}}$

Step 2.3.2.9

Subtract $12 i$ from $12 i$ .

$\frac{- 6 \sqrt{3} + 2 + 6 i + 2 i \sqrt{3}}{4 + 0 - 36 i^{2}}$

Step 2.3.2.10

Add $4$ and $0$ .

$\frac{- 6 \sqrt{3} + 2 + 6 i + 2 i \sqrt{3}}{4 - 36 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{- 6 \sqrt{3} + 2 + 6 i + 2 i \sqrt{3}}{4 - 36 \cdot - 1}$

Step 2.3.3.2

Multiply $- 36$ by $- 1$ .

$\frac{- 6 \sqrt{3} + 2 + 6 i + 2 i \sqrt{3}}{4 + 36}$

Step 2.3.4

Add $4$ and $36$ .

$\frac{- 6 \sqrt{3} + 2 + 6 i + 2 i \sqrt{3}}{40}$

Step 3

Cancel the common factor of $- 6 \sqrt{3} + 2 + 6 i + 2 i \sqrt{3}$ and $40$ .

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Step 3.1

Factor $2$ out of $- 6 \sqrt{3}$ .

$\frac{2 (- 3 \sqrt{3}) + 2 + 6 i + 2 i \sqrt{3}}{40}$

Step 3.2

Factor $2$ out of $2$ .

$\frac{2 (- 3 \sqrt{3}) + 2 \cdot 1 + 6 i + 2 i \sqrt{3}}{40}$

Step 3.3

Factor $2$ out of $6 i$ .

$\frac{2 (- 3 \sqrt{3}) + 2 \cdot 1 + 2 (3 i) + 2 i \sqrt{3}}{40}$

Step 3.4

Factor $2$ out of $2 (1) + 2 (3 i)$ .

$\frac{2 (- 3 \sqrt{3}) + 2 (1 + 3 i) + 2 i \sqrt{3}}{40}$

Step 3.5

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

$\frac{2 (- 3 \sqrt{3}) + 2 (1 + 3 i) + 2 (i \sqrt{3})}{40}$

Step 3.6

Factor $2$ out of $2 (1 + 3 i) + 2 (i \sqrt{3})$ .

$\frac{2 (- 3 \sqrt{3}) + 2 (1 + 3 i + i \sqrt{3})}{40}$

Step 3.7

Factor $2$ out of $2 (- 3 \sqrt{3}) + 2 (1 + 3 i + i \sqrt{3})$ .

$\frac{2 (- 3 \sqrt{3} + 1 + 3 i + i \sqrt{3})}{40}$

Step 3.8

Cancel the common factors.

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Step 3.8.1

Factor $2$ out of $40$ .

$\frac{2 (- 3 \sqrt{3} + 1 + 3 i + i \sqrt{3})}{2 (20)}$

Step 3.8.2

Cancel the common factor.

$\frac{2 (- 3 \sqrt{3} + 1 + 3 i + i \sqrt{3})}{2 \cdot 20}$

Step 3.8.3

Rewrite the expression.

$\frac{- 3 \sqrt{3} + 1 + 3 i + i \sqrt{3}}{20}$

Step 4

Factor $- 1$ out of $- 3 \sqrt{3}$ .

$\frac{- (3 \sqrt{3}) + 1 + 3 i + i \sqrt{3}}{20}$

Step 5

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- (3 \sqrt{3}) - 1 (- 1) + 3 i + i \sqrt{3}}{20}$

Step 6

Factor $- 1$ out of $- (3 \sqrt{3}) - 1 (- 1)$ .

$\frac{- (3 \sqrt{3} - 1) + 3 i + i \sqrt{3}}{20}$

Step 7

Factor $- 1$ out of $3 i$ .

$\frac{- (3 \sqrt{3} - 1) - (- 3 i) + i \sqrt{3}}{20}$

Step 8

Factor $- 1$ out of $- (3 \sqrt{3} - 1) - (- 3 i)$ .

$\frac{- (3 \sqrt{3} - 1 - 3 i) + i \sqrt{3}}{20}$

Step 9

Factor $- 1$ out of $i \sqrt{3}$ .

$\frac{- (3 \sqrt{3} - 1 - 3 i) - (- i \sqrt{3})}{20}$

Step 10

Factor $- 1$ out of $- (3 \sqrt{3} - 1 - 3 i) - (- i \sqrt{3})$ .

$\frac{- (3 \sqrt{3} - 1 - 3 i - i \sqrt{3})}{20}$

Step 11

Simplify the expression.

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Step 11.1

Rewrite $- (3 \sqrt{3} - 1 - 3 i - i \sqrt{3})$ as $- 1 (3 \sqrt{3} - 1 - 3 i - i \sqrt{3})$ .

$\frac{- 1 (3 \sqrt{3} - 1 - 3 i - i \sqrt{3})}{20}$

Step 11.2

Move the negative in front of the fraction.

$- \frac{3 \sqrt{3} - 1 - 3 i - i \sqrt{3}}{20}$

Step 12

Split the fraction $\frac{3 \sqrt{3} - 1 - 3 i - i \sqrt{3}}{20}$ into two fractions.

$- (\frac{3 \sqrt{3} - 1 - 3 i}{20} + \frac{- i \sqrt{3}}{20})$

Step 13

Split the fraction $\frac{3 \sqrt{3} - 1 - 3 i}{20}$ into two fractions.

$- (\frac{3 \sqrt{3} - 1}{20} + \frac{- 3 i}{20} + \frac{- i \sqrt{3}}{20})$

Step 14

Split the fraction $\frac{3 \sqrt{3} - 1}{20}$ into two fractions.

$- (\frac{3 \sqrt{3}}{20} + \frac{- 1}{20} + \frac{- 3 i}{20} + \frac{- i \sqrt{3}}{20})$

Step 15

Move the negative in front of the fraction.

$- (\frac{3 \sqrt{3}}{20} - \frac{1}{20} + \frac{- 3 i}{20} + \frac{- i \sqrt{3}}{20})$

Step 16

Move the negative in front of the fraction.

$- (\frac{3 \sqrt{3}}{20} - \frac{1}{20} - \frac{3 i}{20} + \frac{- i \sqrt{3}}{20})$

Step 17

Move the negative in front of the fraction.

$- (\frac{3 \sqrt{3}}{20} - \frac{1}{20} - \frac{3 i}{20} - \frac{i \sqrt{3}}{20})$