# How do I multiply complex numbers?

Jul 21, 2018

Hope this helps.

#### Explanation:

Step 1: Distribute (or FOIL) to remove the parenthesis.
Step 2 : Simplify the powers of i, specifically remember that i^2 = –1.
Step 3 : Combine like terms, that is, combine real numbers with real numbers and imaginary numbers with imaginary numbers.

Example 1 – Multiply: (4 – 3i)(2 + 5i)

Step 1:
"Distribute to remove the parenthesis. " color(green)(8 + 20 i - 6 i - 15 i^2

Step 2:
"Simplify the powers of i, specifically remember that " i^2 = –1
color(blue)(8 + 20 i - 6 i + 15.

Step 3:
$\text{Combine like terms, that is, combine real numbers with real numbers and imaginary numbers with imaginary numbers.}$
color(crimson)(23 + 14 i

Example 2 – Multiply: (7 – 9i)(4 - 6i)

Step 1:
"Distribute to remove the parenthesis. " color(green)(28 - 42 i - 36 i + 54 i^2

Step 2:
"Simplify the powers of i, specifically remember that " i^2 = –1
color(blue)(28 - 42 i - 36 i - 54.

Step 3:
$\text{Combine like terms, that is, combine real numbers with real numbers and imaginary numbers with imaginary numbers.}$
color(crimson)(-26 - 78 i

Example 3 – Multiply: (7 - 9i)(4 - 6i)(2 + 5i)

"Step 1: Distribute (or FOIL) using only the first two complex numbers " color(green)((28 - 42 i - 36 i + 54 i^2) * (2 + 5 i)

$\text{Step 2: Simplify the powers of i, specifically remember that i2 = –1} \textcolor{b l u e}{28 - 42 i - 36 i - 54} \left(2 + 5 i\right)$

$\text{Step 3: Combine like terms, that is, combine real numbers with real numbers and imaginary numbers with imaginary numbers.}$
color(crimson)((-26 - 78 i) * (2 + 5 i)

"Step 4 : Distribute to remove the parenthesis. " color(chocolate)(-52 - 130 i - 156 i - 390 i^2

"Step 5: Simplify the powers of i, specifically remember that " i^2 = –1
color(magenta)(-52 - 130 i - 156 i + 390.

$\text{Step 6: Combine like terms, that is, combine real numbers with real numbers and imaginary numbers with imaginary numbers.}$
color(purple)(328 - 286 i