Simplify (2+3i)(5-4i)

$(2 + 3 i) (5 - 4 i)$

Step 1

Expand $(2 + 3 i) (5 - 4 i)$ using the FOIL Method.

Apply the distributive property.

$2 (5 - 4 i) + 3 i (5 - 4 i)$

Apply the distributive property.

$2 \cdot 5 + 2 (- 4 i) + 3 i (5 - 4 i)$

Apply the distributive property.

$2 \cdot 5 + 2 (- 4 i) + 3 i \cdot 5 + 3 i (- 4 i)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $5$ .

$10 + 2 (- 4 i) + 3 i \cdot 5 + 3 i (- 4 i)$

Multiply $- 4$ by $2$ .

$10 - 8 i + 3 i \cdot 5 + 3 i (- 4 i)$

Multiply $5$ by $3$ .

$10 - 8 i + 15 i + 3 i (- 4 i)$

Multiply $3 i (- 4 i)$ .

Multiply $- 4$ by $3$ .

$10 - 8 i + 15 i - 12 i i$

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

$10 - 8 i + 15 i - 12 (i^{1} i)$

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

$10 - 8 i + 15 i - 12 (i^{1} i^{1})$

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

$10 - 8 i + 15 i - 12 i^{1 + 1}$

Add $1$ and $1$ .

$10 - 8 i + 15 i - 12 i^{2}$

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

$10 - 8 i + 15 i - 12 \cdot - 1$

Multiply $- 12$ by $- 1$ .

$10 - 8 i + 15 i + 12$

Add $10$ and $12$ .

$22 - 8 i + 15 i$

Add $- 8 i$ and $15 i$ .

$22 + 7 i$