Simplify (4+5i)(4-5i)

$(4 + 5 i) (4 - 5 i)$

Step 1

Expand $(4 + 5 i) (4 - 5 i)$ using the FOIL Method.

Apply the distributive property.

$4 (4 - 5 i) + 5 i (4 - 5 i)$

Apply the distributive property.

$4 \cdot 4 + 4 (- 5 i) + 5 i (4 - 5 i)$

Apply the distributive property.

$4 \cdot 4 + 4 (- 5 i) + 5 i \cdot 4 + 5 i (- 5 i)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$16 + 4 (- 5 i) + 5 i \cdot 4 + 5 i (- 5 i)$

Multiply $- 5$ by $4$ .

$16 - 20 i + 5 i \cdot 4 + 5 i (- 5 i)$

Multiply $4$ by $5$ .

$16 - 20 i + 20 i + 5 i (- 5 i)$

Multiply $5 i (- 5 i)$ .

Multiply $- 5$ by $5$ .

$16 - 20 i + 20 i - 25 i i$

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

$16 - 20 i + 20 i - 25 (i^{1} i)$

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

$16 - 20 i + 20 i - 25 (i^{1} i^{1})$

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

$16 - 20 i + 20 i - 25 i^{1 + 1}$

Add $1$ and $1$ .

$16 - 20 i + 20 i - 25 i^{2}$

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

$16 - 20 i + 20 i - 25 \cdot - 1$

Multiply $- 25$ by $- 1$ .

$16 - 20 i + 20 i + 25$

Add $16$ and $25$ .

$41 - 20 i + 20 i$

Add $- 20 i$ and $20 i$ .

$41 + 0$

Add $41$ and $0$ .

$41$