Simplify (9+2i)(9-2i)

$(9 + 2 i) (9 - 2 i)$

Step 1

Expand $(9 + 2 i) (9 - 2 i)$ using the FOIL Method.

Apply the distributive property.

$9 (9 - 2 i) + 2 i (9 - 2 i)$

Apply the distributive property.

$9 \cdot 9 + 9 (- 2 i) + 2 i (9 - 2 i)$

Apply the distributive property.

$9 \cdot 9 + 9 (- 2 i) + 2 i \cdot 9 + 2 i (- 2 i)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $9$ by $9$ .

$81 + 9 (- 2 i) + 2 i \cdot 9 + 2 i (- 2 i)$

Multiply $- 2$ by $9$ .

$81 - 18 i + 2 i \cdot 9 + 2 i (- 2 i)$

Multiply $9$ by $2$ .

$81 - 18 i + 18 i + 2 i (- 2 i)$

Multiply $2 i (- 2 i)$ .

Multiply $- 2$ by $2$ .

$81 - 18 i + 18 i - 4 i i$

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

$81 - 18 i + 18 i - 4 (i^{1} i)$

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

$81 - 18 i + 18 i - 4 (i^{1} i^{1})$

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

$81 - 18 i + 18 i - 4 i^{1 + 1}$

Add $1$ and $1$ .

$81 - 18 i + 18 i - 4 i^{2}$

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

$81 - 18 i + 18 i - 4 \cdot - 1$

Multiply $- 4$ by $- 1$ .

$81 - 18 i + 18 i + 4$

Add $81$ and $4$ .

$85 - 18 i + 18 i$

Add $- 18 i$ and $18 i$ .

$85 + 0$

Add $85$ and $0$ .

$85$