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

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions: A Step-by-Step Guide

This article will walk you through the process of simplifying the complex expression: (5 + 2yi)(4 - 3i) - (5 - 2yi)(4 - 3i)

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 (i² = -1).

Simplifying the Expression

1. Expand the products:

   • (5 + 2yi)(4 - 3i):

     • Use the distributive property (FOIL method):
     • (5 * 4) + (5 * -3i) + (2yi * 4) + (2yi * -3i)
     • Simplify: 20 - 15i + 8yi - 6i²
     • Substitute i² with -1: 20 - 15i + 8yi + 6 = 26 - 15i + 8yi

   • (5 - 2yi)(4 - 3i):

     • Follow the same process:
     • (5 * 4) + (5 * -3i) + (-2yi * 4) + (-2yi * -3i)
     • Simplify: 20 - 15i - 8yi + 6i²
     • Substitute i² with -1: 20 - 15i - 8yi - 6 = 14 - 15i - 8yi

2. Subtract the expanded expressions:

   • (26 - 15i + 8yi) - (14 - 15i - 8yi)

3. Combine like terms:

   • (26 - 14) + (-15i + 15i) + (8yi + 8yi) = 12 + 16yi

Final Result

The simplified expression is 12 + 16yi.