Examples

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

Step 1

Multiply the numerator and denominator of $\frac{- 4}{3 + 1 i}$ by the conjugate of $3 + 1 i$ to make the denominator real.

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

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Multiply $- 4$ by $3$ .

$\frac{- 12 - 4 (- i)}{(3 + 1 i) (3 - i)}$

Step 2.2.3

Multiply $- 1$ by $- 4$ .

$\frac{- 12 + 4 i}{(3 + 1 i) (3 - i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(3 + 1 i) (3 - i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{- 12 + 4 i}{3 (3 - i) + 1 i (3 - i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 12 + 4 i}{3 \cdot 3 + 3 (- i) + 1 i (3 - i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 12 + 4 i}{3 \cdot 3 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $3$ by $3$ .

$\frac{- 12 + 4 i}{9 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}$

Step 2.3.2.2

Multiply $- 1$ by $3$ .

$\frac{- 12 + 4 i}{9 - 3 i + 1 i \cdot 3 + 1 i (- i)}$

Step 2.3.2.3

Multiply $3$ by $1$ .

$\frac{- 12 + 4 i}{9 - 3 i + 3 i + 1 i (- i)}$

Step 2.3.2.4

Multiply $- 1$ by $1$ .

$\frac{- 12 + 4 i}{9 - 3 i + 3 i - i i}$

Step 2.3.2.5

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

$\frac{- 12 + 4 i}{9 - 3 i + 3 i - (i^{1} i)}$

Step 2.3.2.6

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

$\frac{- 12 + 4 i}{9 - 3 i + 3 i - (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{- 12 + 4 i}{9 - 3 i + 3 i - i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{- 12 + 4 i}{9 - 3 i + 3 i - i^{2}}$

Step 2.3.2.9

Add $- 3 i$ and $3 i$ .

$\frac{- 12 + 4 i}{9 + 0 - i^{2}}$

Step 2.3.2.10

Add $9$ and $0$ .

$\frac{- 12 + 4 i}{9 - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{- 12 + 4 i}{9 - - 1}$

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{- 12 + 4 i}{9 + 1}$

Step 2.3.4

Add $9$ and $1$ .

$\frac{- 12 + 4 i}{10}$

Step 3

Cancel the common factor of $- 12 + 4 i$ and $10$ .

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Step 3.1

Factor $2$ out of $- 12$ .

$\frac{2 (- 6) + 4 i}{10}$

Step 3.2

Factor $2$ out of $4 i$ .

$\frac{2 (- 6) + 2 (2 i)}{10}$

Step 3.3

Factor $2$ out of $2 (- 6) + 2 (2 i)$ .

$\frac{2 (- 6 + 2 i)}{10}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $2$ out of $10$ .

$\frac{2 (- 6 + 2 i)}{2 \cdot 5}$

Step 3.4.2

Cancel the common factor.

$\frac{2 (- 6 + 2 i)}{2 \cdot 5}$

Step 3.4.3

Rewrite the expression.

$\frac{- 6 + 2 i}{5}$

Step 4

Split the fraction $\frac{- 6 + 2 i}{5}$ into two fractions.

$\frac{- 6}{5} + \frac{2 i}{5}$

Step 5

Move the negative in front of the fraction.

$- \frac{6}{5} + \frac{2 i}{5}$

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