Simplify (1-i)/(-2+2i)

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

Multiply the numerator and denominator of $\frac{1 - i}{- 2 + 2 i}$ by the conjugate of $- 2 + 2 i$ to make the denominator real.

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

Multiply.

Combine.

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

Simplify the numerator.

Expand $(1 - i) (- 2 - 2 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $- 2$ by $1$ .

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

Multiply $- 2 i$ by $1$ .

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

Multiply $- 2$ by $- 1$ .

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

Multiply $- i (- 2 i)$ .

Multiply $- 2$ by $- 1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $2$ by $- 1$ .

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

Subtract $2$ from $- 2$ .

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

Add $- 2 i$ and $2 i$ .

$\frac{- 4 + 0}{(- 2 + 2 i) (- 2 - 2 i)}$

Add $- 4$ and $0$ .

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $- 2$ by $- 2$ .

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

Multiply $- 2$ by $- 2$ .

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

Multiply $- 2$ by $2$ .

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

Multiply $- 2$ by $2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Subtract $4 i$ from $4 i$ .

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

Add $4$ and $0$ .

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

Simplify each term.

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

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

Multiply $- 4$ by $- 1$ .

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

Add $4$ and $4$ .

$\frac{- 4}{8}$

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

Factor $4$ out of $- 4$ .

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

Cancel the common factors.

Factor $4$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{- 1}{2}$

Move the negative in front of the fraction.

$- \frac{1}{2}$

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$

Simplify (1-i)/(-2+2i)

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