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

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

Step 1

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$4 + 2 (- 9 i) + 9 i \cdot 2 + 9 i (- 9 i)$

Multiply $- 9$ by $2$ .

$4 - 18 i + 9 i \cdot 2 + 9 i (- 9 i)$

Multiply $2$ by $9$ .

$4 - 18 i + 18 i + 9 i (- 9 i)$

Multiply $9 i (- 9 i)$ .

Multiply $- 9$ by $9$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 81$ by $- 1$ .

$4 - 18 i + 18 i + 81$

Add $4$ and $81$ .

$85 - 18 i + 18 i$

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

$85 + 0$

Add $85$ and $0$ .

$85$