# How do I find the product of two imaginary numbers?

Sep 4, 2014

First, complex numbers can come in a variety of forms!

Ex: multiply $3 i \cdot - 4 i =$

Remember, with multiplication you can rearrange the order (called the Commutative Property):

$3 \cdot - 4 \cdot i \cdot i = - 12 {i}^{2}$

... and then always substitute -1 for ${i}^{2}$:

$- 12 \cdot - 1 = 12$

Ex: the numbers might come in a radical form:

$\sqrt{- 3} \cdot 4 \sqrt{- 12} =$

You should always "factor" out the imaginary part from the square roots like this:

$\sqrt{- 1} \sqrt{3} \cdot 4 \cdot \sqrt{- 1} \sqrt{4} \sqrt{3} =$

and simplify again:

$= i \cdot 4 \cdot \sqrt{3} \cdot \sqrt{3} \cdot \sqrt{4}$
$= i \cdot 4 \cdot 3 \cdot 2 = 24 i$

Ex: what about the Distributive Property? $3 i \left(4 i - 6\right) =$

$= 12 {i}^{2} - 18 i$
$= 12 \left(- 1\right) - 18 i$
$= - 12 - 18 i$

And last but not least, a pair of binomials in a + bi form:

Ex: (3 - 2i)(4 + i) =

=12 + 3i - 8i - $2 {i}^{2}$
= 12 - 2(-1) + 3i - 8i
= 12 + 2 - 5i
= 14 - 5i