Simplify (9+5i)(9-5i)

$(9 + 5 i) (9 - 5 i)$

Step 1

Expand $(9 + 5 i) (9 - 5 i)$ using the FOIL Method.

Apply the distributive property.

$9 (9 - 5 i) + 5 i (9 - 5 i)$

Apply the distributive property.

$9 \cdot 9 + 9 (- 5 i) + 5 i (9 - 5 i)$

Apply the distributive property.

$9 \cdot 9 + 9 (- 5 i) + 5 i \cdot 9 + 5 i (- 5 i)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $9$ by $9$ .

$81 + 9 (- 5 i) + 5 i \cdot 9 + 5 i (- 5 i)$

Multiply $- 5$ by $9$ .

$81 - 45 i + 5 i \cdot 9 + 5 i (- 5 i)$

Multiply $9$ by $5$ .

$81 - 45 i + 45 i + 5 i (- 5 i)$

Multiply $5 i (- 5 i)$ .

Multiply $- 5$ by $5$ .

$81 - 45 i + 45 i - 25 i i$

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

$81 - 45 i + 45 i - 25 (i^{1} i)$

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

$81 - 45 i + 45 i - 25 (i^{1} i^{1})$

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

$81 - 45 i + 45 i - 25 i^{1 + 1}$

Add $1$ and $1$ .

$81 - 45 i + 45 i - 25 i^{2}$

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

$81 - 45 i + 45 i - 25 \cdot - 1$

Multiply $- 25$ by $- 1$ .

$81 - 45 i + 45 i + 25$

Add $81$ and $25$ .

$106 - 45 i + 45 i$

Add $- 45 i$ and $45 i$ .

$106 + 0$

Add $106$ and $0$ .

$106$