# Multiply Complex Numbers

How to multiply complex numbers: 2 examples and their solutions.

## Example(3 + 2i)(1 + 6i)

Solution

Solution (Detail)

Multiply the outer terms.

3⋅(+6i) = +18i

Multiply the inner terms.

+2i⋅1 = +2i

Multiply the last two terms.

+2i⋅(+6i)

= +12⋅i^{2}

= +12⋅(-1)

= -12

Power of i

So

(3 + 2i)(1 + 6i)

= 3 + 18i + 2i - 12.

3 - 12 = -9

+18i + 2i = +20i

Add and Subtract Complex Numbers

So

-9 + 20i

is the answer.

## Example(4 + i)(8 - 5i)

Solution

Solution (Detail)

Use the FOIL method

to multiply these two complex numbers.

Multiply the first two terms.

4⋅8 = 32

Multiply the outer terms.

4⋅(-5i) = -20i

Multiply the inner terms.

+i⋅8 = +8i

Multiply the last two terms.

+i⋅(-5i)

= -5⋅i^{2}

= -5⋅(-1)

= +5

So

(4 + i)(8 - 5i)

= 32 - 20i + 8i + 5.

32 + 5 = 37

-20i + 8i = -12i

So

37 - 12i

is the answer.