(5/-3+2i+2/1-i)(4-5i/3+2i)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions: A Step-by-Step Guide

This article aims to demonstrate the process of simplifying complex expressions, focusing on the specific example: (5/-3+2i+2/1-i)(4-5i/3+2i)

Understanding the Problem

The expression involves complex numbers, represented by the imaginary unit i (where i² = -1). Simplifying this expression requires understanding the rules of complex number arithmetic and applying them to each individual operation.

Breaking Down the Expression

Before jumping into the calculations, let's break down the expression into smaller, more manageable parts:

1. Simplify the fractions:

   • (5/-3 + 2i) = -5/3 + 2i
   • (2/1-i) = 2/(1-i) * (1+i)/(1+i) = 2(1+i)/(1+1) = 1+i
   • (4-5i/3+2i) = (4-5i)/(3+2i) * (3-2i)/(3-2i) = (12 - 10i - 15i + 10i²)/(9 + 4) = (2 - 25i)/13

2. Re-write the expression:

   • The original expression now becomes: (-5/3 + 2i + 1 + i) * (2 - 25i)/13

Simplifying the Expression

Now, we can focus on simplifying the expression step-by-step:

1. Combine real and imaginary terms:

   • (-5/3 + 2i + 1 + i) = (-2/3 + 3i)

2. Multiply the two complex numbers:

   • (-2/3 + 3i) * (2 - 25i)/13 = [(-2/3 * 2) + (-2/3 * -25i) + (3i * 2) + (3i * -25i)] / 13
   • = [-4/3 + 50i/3 + 6i - 75i²] / 13

3. Remember i² = -1:

   • = [-4/3 + 50i/3 + 6i + 75] / 13

4. Combine real and imaginary terms:

   • = [(221/3) + (62i/3)] / 13

5. Simplify the fraction:

   • = 221/39 + 62i/39

Conclusion

Therefore, the simplified form of the complex expression (5/-3+2i+2/1-i)(4-5i/3+2i) is 221/39 + 62i/39. This process showcases the importance of understanding complex number operations and applying them systematically to simplify complex expressions.