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

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

Step 1

Cancel the common factor of $1 - i$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 2

Multiply $\frac{1}{1 + i}$ by $1$ .

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

Step 3

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

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

Step 4

Multiply.

Combine.

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

Multiply $1 - i$ by $1$ .

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $1$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

Multiply $1$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Add $- 1 i$ and $1 i$ .

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

Add $1$ and $0$ .

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

Simplify each term.

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

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

Multiply $- 1$ by $- 1$ .

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

Add $1$ and $1$ .

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

Step 5

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

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

Step 6

Move the negative in front of the fraction.

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

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

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