Simplify (5+3i)(2-3i)

$(5 + 3 i) (2 - 3 i)$

Step 1

Expand $(5 + 3 i) (2 - 3 i)$ using the FOIL Method.

Apply the distributive property.

$5 (2 - 3 i) + 3 i (2 - 3 i)$

Apply the distributive property.

$5 \cdot 2 + 5 (- 3 i) + 3 i (2 - 3 i)$

Apply the distributive property.

$5 \cdot 2 + 5 (- 3 i) + 3 i \cdot 2 + 3 i (- 3 i)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $2$ .

$10 + 5 (- 3 i) + 3 i \cdot 2 + 3 i (- 3 i)$

Multiply $- 3$ by $5$ .

$10 - 15 i + 3 i \cdot 2 + 3 i (- 3 i)$

Multiply $2$ by $3$ .

$10 - 15 i + 6 i + 3 i (- 3 i)$

Multiply $3 i (- 3 i)$ .

Multiply $- 3$ by $3$ .

$10 - 15 i + 6 i - 9 i i$

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

$10 - 15 i + 6 i - 9 (i^{1} i)$

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

$10 - 15 i + 6 i - 9 (i^{1} i^{1})$

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

$10 - 15 i + 6 i - 9 i^{1 + 1}$

Add $1$ and $1$ .

$10 - 15 i + 6 i - 9 i^{2}$

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

$10 - 15 i + 6 i - 9 \cdot - 1$

Multiply $- 9$ by $- 1$ .

$10 - 15 i + 6 i + 9$

Add $10$ and $9$ .

$19 - 15 i + 6 i$

Add $- 15 i$ and $6 i$ .

$19 - 9 i$