Divide (2i)/(1-i)

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $2$ by $1$ .

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

Multiply $2 i i$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

Simplify each term.

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

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

Multiply $2$ by $- 1$ .

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

Reorder $2 i$ and $- 2$ .

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $1$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Subtract $i$ from $1 i$ .

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

Add $1$ and $0$ .

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

Simplify each term.

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

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

Multiply $- 1$ by $- 1$ .

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

Add $1$ and $1$ .

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

Step 3

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

Factor $2$ out of $- 2$ .

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

Factor $2$ out of $2 i$ .

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

Factor $2$ out of $2 (- 1) + 2 (i)$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

$- 1 + i$

Divide (2i)/(1-i)

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