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

$(1 + 2 i) (1 - 2 i)$

Step 1

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

Apply the distributive property.

$1 (1 - 2 i) + 2 i (1 - 2 i)$

Apply the distributive property.

$1 \cdot 1 + 1 (- 2 i) + 2 i (1 - 2 i)$

Apply the distributive property.

$1 \cdot 1 + 1 (- 2 i) + 2 i \cdot 1 + 2 i (- 2 i)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$1 + 1 (- 2 i) + 2 i \cdot 1 + 2 i (- 2 i)$

Multiply $- 2 i$ by $1$ .

$1 - 2 i + 2 i \cdot 1 + 2 i (- 2 i)$

Multiply $2$ by $1$ .

$1 - 2 i + 2 i + 2 i (- 2 i)$

Multiply $2 i (- 2 i)$ .

Multiply $- 2$ by $2$ .

$1 - 2 i + 2 i - 4 i i$

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

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

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

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

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

$1 - 2 i + 2 i - 4 i^{1 + 1}$

Add $1$ and $1$ .

$1 - 2 i + 2 i - 4 i^{2}$

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

$1 - 2 i + 2 i - 4 \cdot - 1$

Multiply $- 4$ by $- 1$ .

$1 - 2 i + 2 i + 4$

Add $1$ and $4$ .

$5 - 2 i + 2 i$

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

$5 + 0$

Add $5$ and $0$ .

$5$