Find the Complex Conjugate (3+2i)(3-2i)

$(3 + 2 i) (3 - 2 i)$

Step 1

Expand $(3 + 2 i) (3 - 2 i)$ using the FOIL Method.

Apply the distributive property.

$3 (3 - 2 i) + 2 i (3 - 2 i)$

Apply the distributive property.

$3 \cdot 3 + 3 (- 2 i) + 2 i (3 - 2 i)$

Apply the distributive property.

$3 \cdot 3 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$9 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)$

Multiply $- 2$ by $3$ .

$9 - 6 i + 2 i \cdot 3 + 2 i (- 2 i)$

Multiply $3$ by $2$ .

$9 - 6 i + 6 i + 2 i (- 2 i)$

Multiply $2 i (- 2 i)$ .

Multiply $- 2$ by $2$ .

$9 - 6 i + 6 i - 4 i i$

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

$9 - 6 i + 6 i - 4 (i^{1} i)$

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

$9 - 6 i + 6 i - 4 (i^{1} i^{1})$

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

$9 - 6 i + 6 i - 4 i^{1 + 1}$

Add $1$ and $1$ .

$9 - 6 i + 6 i - 4 i^{2}$

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

$9 - 6 i + 6 i - 4 \cdot - 1$

Multiply $- 4$ by $- 1$ .

$9 - 6 i + 6 i + 4$

Add $9$ and $4$ .

$13 - 6 i + 6 i$

Add $- 6 i$ and $6 i$ .

$13 + 0$

Add $13$ and $0$ .

$13$

Step 3

Since there are no imaginary terms, the complex conjugate is the same as the simplified expression.

$13$