(4-i)(-3+7i)-7i(8+2i)

Jasonbradley

2 min read

·

Jun 16, 2024

21

Share

Simplifying Complex Numbers: A Step-by-Step Guide

This article will guide you through the process of simplifying the complex number expression: (4 - i)(-3 + 7i) - 7i(8 + 2i)

Understanding Complex Numbers

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

Simplifying the Expression

To simplify the given expression, we'll use the distributive property and the fact that i² = -1:

1. Expand the products:

   • (4 - i)(-3 + 7i) = 4(-3) + 4(7i) - i(-3) - i(7i) = -12 + 28i + 3i - 7i²
   • 7i(8 + 2i) = 7i(8) + 7i(2i) = 56i + 14i²

2. Substitute i² with -1:

   • -12 + 28i + 3i - 7i² = -12 + 28i + 3i - 7(-1) = -5 + 31i
   • 56i + 14i² = 56i + 14(-1) = -14 + 56i

3. Combine the terms:

   • (-5 + 31i) - (-14 + 56i) = -5 + 31i + 14 - 56i = 9 - 25i

Final Result

Therefore, the simplified form of the complex number expression (4 - i)(-3 + 7i) - 7i(8 + 2i) is 9 - 25i.