Divide (5-8i)/(3+2i)

$\frac{5 - 8 i}{3 + 2 i}$

Step 1

Multiply the numerator and denominator of $\frac{5 - 8 i}{3 + 2 i}$ by the conjugate of $3 + 2 i$ to make the denominator real.

$\frac{5 - 8 i}{3 + 2 i} \cdot \frac{3 - 2 i}{3 - 2 i}$

Step 2

Multiply.

Combine.

$\frac{(5 - 8 i) (3 - 2 i)}{(3 + 2 i) (3 - 2 i)}$

Simplify the numerator.

Expand $(5 - 8 i) (3 - 2 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{5 (3 - 2 i) - 8 i (3 - 2 i)}{(3 + 2 i) (3 - 2 i)}$

Apply the distributive property.

$\frac{5 \cdot 3 + 5 (- 2 i) - 8 i (3 - 2 i)}{(3 + 2 i) (3 - 2 i)}$

Apply the distributive property.

$\frac{5 \cdot 3 + 5 (- 2 i) - 8 i \cdot 3 - 8 i (- 2 i)}{(3 + 2 i) (3 - 2 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $3$ .

$\frac{15 + 5 (- 2 i) - 8 i \cdot 3 - 8 i (- 2 i)}{(3 + 2 i) (3 - 2 i)}$

Multiply $- 2$ by $5$ .

$\frac{15 - 10 i - 8 i \cdot 3 - 8 i (- 2 i)}{(3 + 2 i) (3 - 2 i)}$

Multiply $3$ by $- 8$ .

$\frac{15 - 10 i - 24 i - 8 i (- 2 i)}{(3 + 2 i) (3 - 2 i)}$

Multiply $- 8 i (- 2 i)$ .

Multiply $- 2$ by $- 8$ .

$\frac{15 - 10 i - 24 i + 16 i i}{(3 + 2 i) (3 - 2 i)}$

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

$\frac{15 - 10 i - 24 i + 16 (i^{1} i)}{(3 + 2 i) (3 - 2 i)}$

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

$\frac{15 - 10 i - 24 i + 16 (i^{1} i^{1})}{(3 + 2 i) (3 - 2 i)}$

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

$\frac{15 - 10 i - 24 i + 16 i^{1 + 1}}{(3 + 2 i) (3 - 2 i)}$

Add $1$ and $1$ .

$\frac{15 - 10 i - 24 i + 16 i^{2}}{(3 + 2 i) (3 - 2 i)}$

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

$\frac{15 - 10 i - 24 i + 16 \cdot - 1}{(3 + 2 i) (3 - 2 i)}$

Multiply $16$ by $- 1$ .

$\frac{15 - 10 i - 24 i - 16}{(3 + 2 i) (3 - 2 i)}$

Subtract $16$ from $15$ .

$\frac{- 1 - 10 i - 24 i}{(3 + 2 i) (3 - 2 i)}$

Subtract $24 i$ from $- 10 i$ .

$\frac{- 1 - 34 i}{(3 + 2 i) (3 - 2 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{- 1 - 34 i}{3 (3 - 2 i) + 2 i (3 - 2 i)}$

Apply the distributive property.

$\frac{- 1 - 34 i}{3 \cdot 3 + 3 (- 2 i) + 2 i (3 - 2 i)}$

Apply the distributive property.

$\frac{- 1 - 34 i}{3 \cdot 3 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)}$

Simplify.

Multiply $3$ by $3$ .

$\frac{- 1 - 34 i}{9 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)}$

Multiply $- 2$ by $3$ .

$\frac{- 1 - 34 i}{9 - 6 i + 2 i \cdot 3 + 2 i (- 2 i)}$

Multiply $3$ by $2$ .

$\frac{- 1 - 34 i}{9 - 6 i + 6 i + 2 i (- 2 i)}$

Multiply $- 2$ by $2$ .

$\frac{- 1 - 34 i}{9 - 6 i + 6 i - 4 i i}$

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

$\frac{- 1 - 34 i}{9 - 6 i + 6 i - 4 (i^{1} i)}$

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

$\frac{- 1 - 34 i}{9 - 6 i + 6 i - 4 (i^{1} i^{1})}$

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

$\frac{- 1 - 34 i}{9 - 6 i + 6 i - 4 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 1 - 34 i}{9 - 6 i + 6 i - 4 i^{2}}$

Add $- 6 i$ and $6 i$ .

$\frac{- 1 - 34 i}{9 + 0 - 4 i^{2}}$

Add $9$ and $0$ .

$\frac{- 1 - 34 i}{9 - 4 i^{2}}$

Simplify each term.

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

$\frac{- 1 - 34 i}{9 - 4 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$\frac{- 1 - 34 i}{9 + 4}$

Add $9$ and $4$ .

$\frac{- 1 - 34 i}{13}$

Step 3

Split the fraction $\frac{- 1 - 34 i}{13}$ into two fractions.

$\frac{- 1}{13} + \frac{- 34 i}{13}$

Step 4

Simplify each term.

Move the negative in front of the fraction.

$- \frac{1}{13} + \frac{- 34 i}{13}$

Move the negative in front of the fraction.

$- \frac{1}{13} - \frac{34 i}{13}$

Divide (5-8i)/(3+2i)

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