Simplify (3+5i)(5-3i)

$(3 + 5 i) (5 - 3 i)$

Step 1

Expand $(3 + 5 i) (5 - 3 i)$ using the FOIL Method.

Apply the distributive property.

$3 (5 - 3 i) + 5 i (5 - 3 i)$

Apply the distributive property.

$3 \cdot 5 + 3 (- 3 i) + 5 i (5 - 3 i)$

Apply the distributive property.

$3 \cdot 5 + 3 (- 3 i) + 5 i \cdot 5 + 5 i (- 3 i)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $5$ .

$15 + 3 (- 3 i) + 5 i \cdot 5 + 5 i (- 3 i)$

Multiply $- 3$ by $3$ .

$15 - 9 i + 5 i \cdot 5 + 5 i (- 3 i)$

Multiply $5$ by $5$ .

$15 - 9 i + 25 i + 5 i (- 3 i)$

Multiply $5 i (- 3 i)$ .

Multiply $- 3$ by $5$ .

$15 - 9 i + 25 i - 15 i i$

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

$15 - 9 i + 25 i - 15 (i^{1} i)$

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

$15 - 9 i + 25 i - 15 (i^{1} i^{1})$

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

$15 - 9 i + 25 i - 15 i^{1 + 1}$

Add $1$ and $1$ .

$15 - 9 i + 25 i - 15 i^{2}$

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

$15 - 9 i + 25 i - 15 \cdot - 1$

Multiply $- 15$ by $- 1$ .

$15 - 9 i + 25 i + 15$

Add $15$ and $15$ .

$30 - 9 i + 25 i$

Add $- 9 i$ and $25 i$ .

$30 + 16 i$