Precalculus Examples

$\frac{3 + 5 i}{- 2 + 2 i}$

Step 1

Multiply the numerator and denominator of $\frac{3 + 5 i}{- 2 + 2 i}$ by the conjugate of $- 2 + 2 i$ to make the denominator real.

$\frac{3 + 5 i}{- 2 + 2 i} \cdot \frac{- 2 - 2 i}{- 2 - 2 i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{(3 + 5 i) (- 2 - 2 i)}{(- 2 + 2 i) (- 2 - 2 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand $(3 + 5 i) (- 2 - 2 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

$\frac{3 (- 2 - 2 i) + 5 i (- 2 - 2 i)}{(- 2 + 2 i) (- 2 - 2 i)}$

Step 2.2.1.2

Apply the distributive property.

$\frac{3 \cdot - 2 + 3 (- 2 i) + 5 i (- 2 - 2 i)}{(- 2 + 2 i) (- 2 - 2 i)}$

Step 2.2.1.3

Apply the distributive property.

$\frac{3 \cdot - 2 + 3 (- 2 i) + 5 i \cdot - 2 + 5 i (- 2 i)}{(- 2 + 2 i) (- 2 - 2 i)}$

Step 2.2.2

Simplify and combine like terms.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $3$ by $- 2$ .

$\frac{- 6 + 3 (- 2 i) + 5 i \cdot - 2 + 5 i (- 2 i)}{(- 2 + 2 i) (- 2 - 2 i)}$

Step 2.2.2.1.2

Multiply $- 2$ by $3$ .

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

Step 2.2.2.1.3

Multiply $- 2$ by $5$ .

$\frac{- 6 - 6 i - 10 i + 5 i (- 2 i)}{(- 2 + 2 i) (- 2 - 2 i)}$

Step 2.2.2.1.4

Multiply $5 i (- 2 i)$ .

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Step 2.2.2.1.4.1

Multiply $- 2$ by $5$ .

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

Step 2.2.2.1.4.2

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

$\frac{- 6 - 6 i - 10 i - 10 (i^{1} i)}{(- 2 + 2 i) (- 2 - 2 i)}$

Step 2.2.2.1.4.3

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

$\frac{- 6 - 6 i - 10 i - 10 (i^{1} i^{1})}{(- 2 + 2 i) (- 2 - 2 i)}$

Step 2.2.2.1.4.4

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

$\frac{- 6 - 6 i - 10 i - 10 i^{1 + 1}}{(- 2 + 2 i) (- 2 - 2 i)}$

Step 2.2.2.1.4.5

Add $1$ and $1$ .

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

Step 2.2.2.1.5

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

$\frac{- 6 - 6 i - 10 i - 10 \cdot - 1}{(- 2 + 2 i) (- 2 - 2 i)}$

Step 2.2.2.1.6

Multiply $- 10$ by $- 1$ .

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

Step 2.2.2.2

Add $- 6$ and $10$ .

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

Step 2.2.2.3

Subtract $10 i$ from $- 6 i$ .

$\frac{4 - 16 i}{(- 2 + 2 i) (- 2 - 2 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

$\frac{4 - 16 i}{- 2 (- 2 - 2 i) + 2 i (- 2 - 2 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{4 - 16 i}{- 2 \cdot - 2 - 2 (- 2 i) + 2 i (- 2 - 2 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{4 - 16 i}{- 2 \cdot - 2 - 2 (- 2 i) + 2 i \cdot - 2 + 2 i (- 2 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $- 2$ by $- 2$ .

$\frac{4 - 16 i}{4 - 2 (- 2 i) + 2 i \cdot - 2 + 2 i (- 2 i)}$

Step 2.3.2.2

Multiply $- 2$ by $- 2$ .

$\frac{4 - 16 i}{4 + 4 i + 2 i \cdot - 2 + 2 i (- 2 i)}$

Step 2.3.2.3

Multiply $- 2$ by $2$ .

$\frac{4 - 16 i}{4 + 4 i - 4 i + 2 i (- 2 i)}$

Step 2.3.2.4

Multiply $- 2$ by $2$ .

$\frac{4 - 16 i}{4 + 4 i - 4 i - 4 i i}$

Step 2.3.2.5

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

$\frac{4 - 16 i}{4 + 4 i - 4 i - 4 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{4 - 16 i}{4 + 4 i - 4 i - 4 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{4 - 16 i}{4 + 4 i - 4 i - 4 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{4 - 16 i}{4 + 4 i - 4 i - 4 i^{2}}$

Step 2.3.2.9

Subtract $4 i$ from $4 i$ .

$\frac{4 - 16 i}{4 + 0 - 4 i^{2}}$

Step 2.3.2.10

Add $4$ and $0$ .

$\frac{4 - 16 i}{4 - 4 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{4 - 16 i}{4 - 4 \cdot - 1}$

Step 2.3.3.2

Multiply $- 4$ by $- 1$ .

$\frac{4 - 16 i}{4 + 4}$

Step 2.3.4

Add $4$ and $4$ .

$\frac{4 - 16 i}{8}$

Step 3

Cancel the common factor of $4 - 16 i$ and $8$ .

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Step 3.1

Factor $4$ out of $4$ .

$\frac{4 (1) - 16 i}{8}$

Step 3.2

Factor $4$ out of $- 16 i$ .

$\frac{4 (1) + 4 (- 4 i)}{8}$

Step 3.3

Factor $4$ out of $4 (1) + 4 (- 4 i)$ .

$\frac{4 (1 - 4 i)}{8}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $4$ out of $8$ .

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

$\frac{1 - 4 i}{2}$

Step 4

Split the fraction $\frac{1 - 4 i}{2}$ into two fractions.

$\frac{1}{2} + \frac{- 4 i}{2}$

Step 5

Cancel the common factor of $- 4$ and $2$ .

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Step 5.1

Factor $2$ out of $- 4 i$ .

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $2$ out of $2$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

$\frac{1}{2} + \frac{- 2 i}{1}$

Step 5.2.4

Divide $- 2 i$ by $1$ .

$\frac{1}{2} - 2 i$