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(-1-7i)(2+i)+(8-2i)

2 min read Jun 16, 2024
(-1-7i)(2+i)+(8-2i)

Simplifying Complex Expressions

This article will guide you through the process of simplifying the complex expression: (-1-7i)(2+i)+(8-2i).

Understanding Complex Numbers

Before we begin, let's quickly recap what complex numbers are:

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

Simplifying the Expression

  1. Expand the product:

    We start by expanding the product of the two complex numbers using the distributive property (FOIL method):

    (-1 - 7i)(2 + i) = (-1)(2) + (-1)(i) + (-7i)(2) + (-7i)(i) = -2 - i - 14i - 7i²

  2. Substitute i² with -1:

    Since i² = -1, we can substitute it into the expression:

    -2 - i - 14i - 7i² = -2 - i - 14i - 7(-1)

  3. Combine real and imaginary terms:

    Combine the real terms and the imaginary terms separately:

    -2 + 7 - i - 14i = 5 - 15i

  4. Add the remaining complex number:

    Finally, add the remaining complex number (8 - 2i):

    5 - 15i + (8 - 2i) = 5 + 8 - 15i - 2i = 13 - 17i

Conclusion

The simplified form of the complex expression (-1 - 7i)(2 + i) + (8 - 2i) is 13 - 17i.

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