Simplify (1+i)^-2

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

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Rewrite ${(1 + i)}^{2}$ as $(1 + i) (1 + i)$ .

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

Step 3

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

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

Multiply $i$ by $1$ .

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

Multiply $i$ by $1$ .

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

Multiply $i i$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Subtract $1$ from $1$ .

$\frac{1}{0 + i + i}$

Add $0$ and $i$ .

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

Add $i$ and $i$ .

$\frac{1}{2 i}$