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

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

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

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

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $2$ .

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

Multiply $2$ by $3$ .

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

Multiply $2$ by $- 1$ .

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

Multiply $- i (2 i)$ .

Multiply $2$ by $- 1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 2$ by $- 1$ .

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

Add $6$ and $2$ .

$\frac{8 + 6 i - 2 i}{(2 - 2 i) (2 + 2 i)}$

Subtract $2 i$ from $6 i$ .

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $2$ by $2$ .

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

Multiply $2$ by $2$ .

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

Multiply $2$ by $- 2$ .

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

Multiply $2$ by $- 2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Subtract $4 i$ from $4 i$ .

$\frac{8 + 4 i}{4 + 0 - 4 i^{2}}$

Add $4$ and $0$ .

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

Simplify each term.

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

$\frac{8 + 4 i}{4 - 4 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$\frac{8 + 4 i}{4 + 4}$

Add $4$ and $4$ .

$\frac{8 + 4 i}{8}$

Cancel the common factor of $8 + 4 i$ and $8$ .

Factor $4$ out of $8$ .

$\frac{4 \cdot 2 + 4 i}{8}$

Factor $4$ out of $4 i$ .

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

Factor $4$ out of $4 \cdot 2 + 4 (i)$ .

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

Cancel the common factors.

Factor $4$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Divide $2$ by $2$ .

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

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

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