Simplifying Complex Expressions: A Guide to Handling -5 - 3i ? 2i 4 - 7i

Understanding and simplifying complex expressions is a fundamental concept in algebra and mathematics. In this article, we will guide you through the process of simplifying a specific expression -5 - 3i ? 2i 4 - 7i. We will break down the steps, explain the reasoning behind each step, and provide a comprehensive solution.

Complex Numbers: An Overview

Before diving into the simplification process, it's essential to have a solid understanding of complex numbers. A complex number is typically written as a bi, where a is the real part and b is the imaginary part. The imaginary unit, denoted as i, is defined as the square root of -1 (i2 -1).

The Expression to Simplify: -5 - 3i ? 2i 4 - 7i

Let's start with the given expression: -5 - 3i ? 2i 4 - 7i. Our goal is to simplify this expression step-by-step.

Step-by-Step Simplification

Remove Parentheses: First, we remove any parentheses as there are no parentheses in this expression. Multiply the Terms: Next, we need to handle the multiplication of the imaginary parts. Specifically, we need to calculate -3i ? 2i: -3i ? 2i -6i2 Recall that i2 -1, so -6i2 -6(-1) 6. Collect Like Terms: Now, we substitute the result from the multiplication into the expression: -5 6 4 - 7i Simplify the Real Part: Combine the real numbers in the expression: -5 6 4 5 Final Expression: The simplified form of the expression is 5 - 7i.

Simplified Form

The simplified form of the given expression is:

y 5 - 7i

Further Examples

Here are a few more examples of simplifying complex expressions:

Example 1: -3 2i ? i - 4i2 7i

-3 2i ? i - 4i2 7i -3 2i2 - 4i2 7i -3 - 2 7i -5 7i

Example 2: 6i - 2i2 5i ? i - 3

6i - 2i2 5i2 - 3 6i - 2(-1) 5(-1) - 3 6i 2 - 5 - 3 6i - 6

Conclusion

Simplifying complex expressions involves careful step-by-step calculations and a solid understanding of the properties of complex numbers. By following the outlined process, you can efficiently and accurately simplify any given complex expression. Practice with multiple examples will further enhance your skills in this area.

Key Terms:

Complex numbers Simplification Algebraic expressions