(8+5i)(3+2i)-(4+i)(4-i)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions

This article will walk through the steps to simplify the complex expression: (8 + 5i)(3 + 2i) - (4 + i)(4 - i).

Understanding Complex Numbers

Before we begin, let's understand the basics of complex numbers:

• Complex numbers are numbers of 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)

Simplifying the Expression

1. Expand the products:

   • (8 + 5i)(3 + 2i) = 24 + 16i + 15i + 10i²
   • (4 + i)(4 - i) = 16 - i²

2. Substitute i² with -1:

   • 24 + 16i + 15i + 10(-1) = 14 + 31i
   • 16 - (-1) = 17

3. Subtract the simplified terms:

   • (14 + 31i) - 17 = -3 + 31i

Final Answer

Therefore, the simplified form of the expression (8 + 5i)(3 + 2i) - (4 + i)(4 - i) is -3 + 31i.