Simplify (3-i)/(4+2i)

$\frac{3 - i}{4 + 2 i}$

Multiply the numerator and denominator of $\frac{3 - i}{4 + 2 i}$ by the conjugate of $4 + 2 i$ to make the denominator real.

$\frac{3 - i}{4 + 2 i} \cdot \frac{4 - 2 i}{4 - 2 i}$

Multiply.

Combine.

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

Simplify the numerator.

Expand $(3 - i) (4 - 2 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $4$ .

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

Multiply $- 2$ by $3$ .

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

Multiply $4$ by $- 1$ .

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

Multiply $- i (- 2 i)$ .

Multiply $- 2$ by $- 1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

$\frac{12 - 6 i - 4 i + 2 \cdot - 1}{(4 + 2 i) (4 - 2 i)}$

Multiply $2$ by $- 1$ .

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

Subtract $2$ from $12$ .

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

Subtract $4 i$ from $- 6 i$ .

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

Simplify the denominator.

Expand $(4 + 2 i) (4 - 2 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $4$ by $4$ .

$\frac{10 - 10 i}{16 + 4 (- 2 i) + 2 i \cdot 4 + 2 i (- 2 i)}$

Multiply $- 2$ by $4$ .

$\frac{10 - 10 i}{16 - 8 i + 2 i \cdot 4 + 2 i (- 2 i)}$

Multiply $4$ by $2$ .

$\frac{10 - 10 i}{16 - 8 i + 8 i + 2 i (- 2 i)}$

Multiply $- 2$ by $2$ .

$\frac{10 - 10 i}{16 - 8 i + 8 i - 4 i i}$

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

$\frac{10 - 10 i}{16 - 8 i + 8 i - 4 (i^{1} i)}$

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

$\frac{10 - 10 i}{16 - 8 i + 8 i - 4 (i^{1} i^{1})}$

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

$\frac{10 - 10 i}{16 - 8 i + 8 i - 4 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{10 - 10 i}{16 - 8 i + 8 i - 4 i^{2}}$

Add $- 8 i$ and $8 i$ .

$\frac{10 - 10 i}{16 + 0 - 4 i^{2}}$

Add $16$ and $0$ .

$\frac{10 - 10 i}{16 - 4 i^{2}}$

Simplify each term.

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

$\frac{10 - 10 i}{16 - 4 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$\frac{10 - 10 i}{16 + 4}$

Add $16$ and $4$ .

$\frac{10 - 10 i}{20}$

Cancel the common factor of $10 - 10 i$ and $20$ .

Factor $10$ out of $10$ .

$\frac{10 (1) - 10 i}{20}$

Factor $10$ out of $- 10 i$ .

$\frac{10 (1) + 10 (- i)}{20}$

Factor $10$ out of $10 (1) + 10 (- i)$ .

$\frac{10 (1 - i)}{20}$

Cancel the common factors.

Factor $10$ out of $20$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Move the negative in front of the fraction.

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

Simplify (3-i)/(4+2i)

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