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:
#"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:
#"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)#
#"Step 2: Simplify the powers of i, specifically remember that i2 = –1" color(blue)(28 - 42 i - 36 i - 54)(2 + 5i)#
#"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#.
#"Step 6: Combine like terms, that is, combine real numbers with real numbers and imaginary numbers with imaginary numbers."#
#color(purple)(328 - 286 i#
