Simplify (5+i)/(-3+2i)

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

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

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

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $- 3$ .

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

Multiply $- 2$ by $5$ .

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

Move $- 3$ to the left of $i$ .

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

Multiply $i (- 2 i)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 2$ by $- 1$ .

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

Add $- 15$ and $2$ .

$\frac{- 13 - 10 i - 3 i}{(- 3 + 2 i) (- 3 - 2 i)}$

Subtract $3 i$ from $- 10 i$ .

$\frac{- 13 - 13 i}{(- 3 + 2 i) (- 3 - 2 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{- 13 - 13 i}{- 3 (- 3 - 2 i) + 2 i (- 3 - 2 i)}$

Apply the distributive property.

$\frac{- 13 - 13 i}{- 3 \cdot - 3 - 3 (- 2 i) + 2 i (- 3 - 2 i)}$

Apply the distributive property.

$\frac{- 13 - 13 i}{- 3 \cdot - 3 - 3 (- 2 i) + 2 i \cdot - 3 + 2 i (- 2 i)}$

Simplify.

Multiply $- 3$ by $- 3$ .

$\frac{- 13 - 13 i}{9 - 3 (- 2 i) + 2 i \cdot - 3 + 2 i (- 2 i)}$

Multiply $- 2$ by $- 3$ .

$\frac{- 13 - 13 i}{9 + 6 i + 2 i \cdot - 3 + 2 i (- 2 i)}$

Multiply $- 3$ by $2$ .

$\frac{- 13 - 13 i}{9 + 6 i - 6 i + 2 i (- 2 i)}$

Multiply $- 2$ by $2$ .

$\frac{- 13 - 13 i}{9 + 6 i - 6 i - 4 i i}$

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

$\frac{- 13 - 13 i}{9 + 6 i - 6 i - 4 (i^{1} i)}$

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

$\frac{- 13 - 13 i}{9 + 6 i - 6 i - 4 (i^{1} i^{1})}$

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

$\frac{- 13 - 13 i}{9 + 6 i - 6 i - 4 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 13 - 13 i}{9 + 6 i - 6 i - 4 i^{2}}$

Subtract $6 i$ from $6 i$ .

$\frac{- 13 - 13 i}{9 + 0 - 4 i^{2}}$

Add $9$ and $0$ .

$\frac{- 13 - 13 i}{9 - 4 i^{2}}$

Simplify each term.

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

$\frac{- 13 - 13 i}{9 - 4 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$\frac{- 13 - 13 i}{9 + 4}$

Add $9$ and $4$ .

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

Cancel the common factor of $- 13 - 13 i$ and $13$ .

Factor $13$ out of $- 13$ .

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

Factor $13$ out of $- 13 i$ .

$\frac{13 \cdot - 1 + 13 (- i)}{13}$

Factor $13$ out of $13 (- 1) + 13 (- i)$ .

$\frac{13 (- 1 - i)}{13}$

Cancel the common factors.

Factor $13$ out of $13$ .

$\frac{13 (- 1 - i)}{13 (1)}$

Cancel the common factor.

$\frac{13 (- 1 - i)}{13 \cdot 1}$

Rewrite the expression.

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

Divide $- 1 - i$ by $1$ .

$- 1 - i$

Simplify (5+i)/(-3+2i)

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