(-7i)(-2+7i)(2+6i)

Jasonbradley

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Jun 16, 2024

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Multiplying Complex Numbers: A Step-by-Step Guide

This article will guide you through the process of multiplying complex numbers. We will work with the expression (-7i)(-2 + 7i)(2 + 6i).

Understanding Complex Numbers

Complex numbers are numbers that consist of a real part and an imaginary part. They are typically written in the form a + bi, where:

• a is the real part
• b is the imaginary part
• i is the imaginary unit, defined as the square root of -1 (i² = -1)

Multiplication of Complex Numbers

To multiply complex numbers, we follow the distributive property just like with regular polynomials:

1. Expand the expression: (-7i)(-2 + 7i)(2 + 6i) = (-7i)[(-2 + 7i)(2 + 6i)]

2. Apply the distributive property to the inner parentheses: (-7i)[(-2 + 7i)(2 + 6i)] = (-7i)[(-2)(2) + (-2)(6i) + (7i)(2) + (7i)(6i)]

3. Simplify the terms: (-7i)[-4 - 12i + 14i + 42i²]

4. Remember that i² = -1, and substitute: (-7i)[-4 - 12i + 14i - 42]

5. Combine like terms: (-7i)[-46 + 2i]

6. Apply the distributive property again: (-7i)(-46) + (-7i)(2i)

7. Simplify: 322i - 14i²

8. Substitute i² = -1: 322i + 14

9. Rearrange to standard form (a + bi): 14 + 322i

Final Answer

Therefore, the product of the complex numbers (-7i)(-2 + 7i)(2 + 6i) is 14 + 322i.