(9-i)(1+3i)(-2-2i)

2 min read Jun 16, 2024

Multiplying Complex Numbers: A Step-by-Step Guide

This article will guide you through multiplying the complex numbers (9 - i)(1 + 3i)(-2 - 2i).

Understanding Complex Numbers

Complex numbers are numbers that can be expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as the square root of -1 (i² = -1).

Multiplication Process

To multiply complex numbers, we use the distributive property (or FOIL method) just like we would with any binomial multiplication.

Step 1: Multiply the first two complex numbers: (9 - i)(1 + 3i)

(9 - i)(1 + 3i) = 9(1) + 9(3i) - i(1) - i(3i)
= 9 + 27i - i - 3i²
= 9 + 26i + 3 (since i² = -1)
= 12 + 26i

Step 2: Multiply the result from Step 1 with the third complex number: (12 + 26i)(-2 - 2i)

(12 + 26i)(-2 - 2i) = 12(-2) + 12(-2i) + 26i(-2) + 26i(-2i)
= -24 - 24i - 52i - 52i²
= -24 - 76i + 52 (since i² = -1)
= 28 - 76i

Final Result

Therefore, the product of (9 - i)(1 + 3i)(-2 - 2i) is 28 - 76i.

Key Points to Remember

Remember that i² = -1
Use the distributive property (or FOIL method) to multiply complex numbers.
Combine real and imaginary terms separately.
Express the final answer in the form a + bi.