Simplify (5+8i)(5-8i)

$(5 + 8 i) (5 - 8 i)$

Step 1

Expand $(5 + 8 i) (5 - 8 i)$ using the FOIL Method.

Apply the distributive property.

$5 (5 - 8 i) + 8 i (5 - 8 i)$

Apply the distributive property.

$5 \cdot 5 + 5 (- 8 i) + 8 i (5 - 8 i)$

Apply the distributive property.

$5 \cdot 5 + 5 (- 8 i) + 8 i \cdot 5 + 8 i (- 8 i)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$25 + 5 (- 8 i) + 8 i \cdot 5 + 8 i (- 8 i)$

Multiply $- 8$ by $5$ .

$25 - 40 i + 8 i \cdot 5 + 8 i (- 8 i)$

Multiply $5$ by $8$ .

$25 - 40 i + 40 i + 8 i (- 8 i)$

Multiply $8 i (- 8 i)$ .

Multiply $- 8$ by $8$ .

$25 - 40 i + 40 i - 64 i i$

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

$25 - 40 i + 40 i - 64 (i^{1} i)$

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

$25 - 40 i + 40 i - 64 (i^{1} i^{1})$

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

$25 - 40 i + 40 i - 64 i^{1 + 1}$

Add $1$ and $1$ .

$25 - 40 i + 40 i - 64 i^{2}$

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

$25 - 40 i + 40 i - 64 \cdot - 1$

Multiply $- 64$ by $- 1$ .

$25 - 40 i + 40 i + 64$

Add $25$ and $64$ .

$89 - 40 i + 40 i$

Add $- 40 i$ and $40 i$ .

$89 + 0$

Add $89$ and $0$ .

$89$