## 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.