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(−2 − 2i)(−4 − 3i)(7 + 8i)

2 min read Jun 17, 2024
(−2 − 2i)(−4 − 3i)(7 + 8i)

Multiplying Complex Numbers: A Step-by-Step Guide

This article will guide you through the process of multiplying three complex numbers: (-2 - 2i)(-4 - 3i)(7 + 8i).

Understanding Complex Numbers

Complex numbers are numbers of the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as the square root of -1.

Multiplication of Complex Numbers

To multiply complex numbers, we use the distributive property, similar to multiplying binomials.

  1. Expand the First Two Factors:

    (-2 - 2i)(-4 - 3i) = (-2)(-4) + (-2)(-3i) + (-2i)(-4) + (-2i)(-3i)

    Simplifying this expression, we get:

    8 + 6i + 8i + 6i²

    Since i² = -1, we can substitute and simplify further:

    8 + 6i + 8i - 6 = 2 + 14i

  2. Multiply the Result by the Third Factor:

    (2 + 14i)(7 + 8i) = 2(7) + 2(8i) + 14i(7) + 14i(8i)

    Expanding and simplifying:

    14 + 16i + 98i + 112i² = 14 + 114i - 112 = -98 + 114i

Conclusion

Therefore, the product of the complex numbers (-2 - 2i)(-4 - 3i)(7 + 8i) is -98 + 114i.

Remember, the key is to apply the distributive property and carefully simplify the terms, remembering that i² = -1.

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