Simplify (2-4i)/(i-1)

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

Multiply the numerator and denominator of $\frac{2 - 4 i}{- 1 + 1 i}$ by the conjugate of $- 1 + 1 i$ to make the denominator real.

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

Multiply.

Combine.

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

Simplify the numerator.

Expand $(2 - 4 i) (- 1 - i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $- 1$ .

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

Multiply $- 1$ by $2$ .

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

Multiply $- 1$ by $- 4$ .

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

Multiply $- 4 i (- i)$ .

Multiply $- 1$ by $- 4$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $4$ by $- 1$ .

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

Subtract $4$ from $- 2$ .

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

Add $- 2 i$ and $4 i$ .

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $- 1$ by $- 1$ .

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

Multiply $- 1$ by $- 1$ .

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

Multiply $- 1$ by $1$ .

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

Multiply $- 1$ by $1$ .

$\frac{- 6 + 2 i}{1 + 1 i - i - i i}$

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

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

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

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

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

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

Add $1$ and $1$ .

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

Subtract $i$ from $1 i$ .

$\frac{- 6 + 2 i}{1 + 0 - i^{2}}$

Add $1$ and $0$ .

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

Simplify each term.

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

$\frac{- 6 + 2 i}{1 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{- 6 + 2 i}{1 + 1}$

Add $1$ and $1$ .

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

Cancel the common factor of $- 6 + 2 i$ and $2$ .

Factor $2$ out of $- 6$ .

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

Factor $2$ out of $2 i$ .

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

Factor $2$ out of $2 (- 3) + 2 (i)$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{- 3 + i}{1}$

Divide $- 3 + i$ by $1$ .

$- 3 + i$

Simplify (2-4i)/(i-1)

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