Simplify (6+9i)(6-9i)

$(6 + 9 i) (6 - 9 i)$

Step 1

Expand $(6 + 9 i) (6 - 9 i)$ using the FOIL Method.

Apply the distributive property.

$6 (6 - 9 i) + 9 i (6 - 9 i)$

Apply the distributive property.

$6 \cdot 6 + 6 (- 9 i) + 9 i (6 - 9 i)$

Apply the distributive property.

$6 \cdot 6 + 6 (- 9 i) + 9 i \cdot 6 + 9 i (- 9 i)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $6$ by $6$ .

$36 + 6 (- 9 i) + 9 i \cdot 6 + 9 i (- 9 i)$

Multiply $- 9$ by $6$ .

$36 - 54 i + 9 i \cdot 6 + 9 i (- 9 i)$

Multiply $6$ by $9$ .

$36 - 54 i + 54 i + 9 i (- 9 i)$

Multiply $9 i (- 9 i)$ .

Multiply $- 9$ by $9$ .

$36 - 54 i + 54 i - 81 i i$

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

$36 - 54 i + 54 i - 81 (i^{1} i)$

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

$36 - 54 i + 54 i - 81 (i^{1} i^{1})$

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

$36 - 54 i + 54 i - 81 i^{1 + 1}$

Add $1$ and $1$ .

$36 - 54 i + 54 i - 81 i^{2}$

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

$36 - 54 i + 54 i - 81 \cdot - 1$

Multiply $- 81$ by $- 1$ .

$36 - 54 i + 54 i + 81$

Add $36$ and $81$ .

$117 - 54 i + 54 i$

Add $- 54 i$ and $54 i$ .

$117 + 0$

Add $117$ and $0$ .

$117$