Simplify (3+2i)(7-5i)

$(3 + 2 i) (7 - 5 i)$

Step 1

Expand $(3 + 2 i) (7 - 5 i)$ using the FOIL Method.

Apply the distributive property.

$3 (7 - 5 i) + 2 i (7 - 5 i)$

Apply the distributive property.

$3 \cdot 7 + 3 (- 5 i) + 2 i (7 - 5 i)$

Apply the distributive property.

$3 \cdot 7 + 3 (- 5 i) + 2 i \cdot 7 + 2 i (- 5 i)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $7$ .

$21 + 3 (- 5 i) + 2 i \cdot 7 + 2 i (- 5 i)$

Multiply $- 5$ by $3$ .

$21 - 15 i + 2 i \cdot 7 + 2 i (- 5 i)$

Multiply $7$ by $2$ .

$21 - 15 i + 14 i + 2 i (- 5 i)$

Multiply $2 i (- 5 i)$ .

Multiply $- 5$ by $2$ .

$21 - 15 i + 14 i - 10 i i$

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

$21 - 15 i + 14 i - 10 (i^{1} i)$

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

$21 - 15 i + 14 i - 10 (i^{1} i^{1})$

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

$21 - 15 i + 14 i - 10 i^{1 + 1}$

Add $1$ and $1$ .

$21 - 15 i + 14 i - 10 i^{2}$

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

$21 - 15 i + 14 i - 10 \cdot - 1$

Multiply $- 10$ by $- 1$ .

$21 - 15 i + 14 i + 10$

Add $21$ and $10$ .

$31 - 15 i + 14 i$

Add $- 15 i$ and $14 i$ .

$31 - i$