Simplify (a+bi)(c+di)

$(a + b i) (c + d i)$

Expand $(a + b i) (c + d i)$ using the FOIL Method.

Apply the distributive property.

$a (c + d i) + b i (c + d i)$

Apply the distributive property.

$a c + a (d i) + b i (c + d i)$

Apply the distributive property.

$a c + a (d i) + b i c + b i (d i)$

Simplify each term.

Rewrite using the commutative property of multiplication.

$a c + i a d + b i c + b i (d i)$

Multiply $b i (d i)$ .

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

$a c + i a d + b i c + b (d (i^{1} i))$

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

$a c + i a d + b i c + b (d (i^{1} i^{1}))$

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

$a c + i a d + b i c + b (d i^{1 + 1})$

Add $1$ and $1$ .

$a c + i a d + b i c + b (d i^{2})$

Rewrite using the commutative property of multiplication.

$a c + i a d + b i c + i^{2} b d$

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

$a c + i a d + b i c - b d$