Solving Complex Number Expressions A Step-by-Step Guide

Introduction

In the realm of mathematics, complex numbers hold a significant position, extending the concept of real numbers by incorporating an imaginary unit, denoted as 'i', which is defined as the square root of -1. This allows us to deal with equations that have no real solutions, opening up a whole new dimension of mathematical possibilities. Understanding operations with complex numbers is crucial in various fields, including electrical engineering, quantum mechanics, and fluid dynamics. This article aims to dissect a specific complex number expression, providing a detailed walkthrough of the solution process to enhance your understanding of these fascinating numbers. We will explore the fundamental principles of complex number arithmetic and apply them to solve the expression: $(8-3i)-(8-3i)(8+8i)$. By the end of this journey, you'll not only grasp the solution but also gain a deeper appreciation for the elegance and power of complex number manipulation. Let's embark on this mathematical adventure together!

Dissecting the Expression: $(8-3i)-(8-3i)(8+8i)$

The expression we aim to solve is $(8-3i)-(8-3i)(8+8i)$. This seemingly simple expression involves complex numbers and requires a careful application of arithmetic operations to arrive at the correct answer. The expression combines subtraction and multiplication of complex numbers, necessitating a systematic approach to ensure accuracy. Our primary objective is to simplify this expression into the standard form of a complex number, which is $a + bi$, where 'a' represents the real part and 'b' represents the imaginary part. To achieve this, we will first tackle the multiplication component, $(8-3i)(8+8i)$, using the distributive property, also known as the FOIL method (First, Outer, Inner, Last). This will help us expand the product of the two complex numbers into a sum of terms. Following the multiplication, we will then perform the subtraction operation, combining like terms (real and imaginary parts) to arrive at the final simplified form. This step-by-step process will not only lead us to the solution but also provide a clear understanding of how complex number operations are performed. This detailed breakdown will be invaluable in tackling similar expressions in the future.

Step-by-Step Solution

1. Multiplication of Complex Numbers: $(8-3i)(8+8i)$

To begin, we focus on the multiplication part of the expression: $(8-3i)(8+8i)$. This involves multiplying two complex numbers, and we'll use the distributive property (FOIL method) to expand the expression. The FOIL method stands for First, Outer, Inner, Last, which dictates the order in which we multiply the terms. First, we multiply the first terms of each complex number: $8 * 8 = 64$. Outer, we multiply the outer terms: $8 * 8i = 64i$. Inner, we multiply the inner terms: $-3i * 8 = -24i$. Last, we multiply the last terms: $-3i * 8i = -24i^2$. Now, we combine these results: $64 + 64i - 24i - 24i^2$. Remember that $i^2$ is equal to -1, so we can substitute this value into the expression: $64 + 64i - 24i - 24(-1)$. This simplifies to $64 + 64i - 24i + 24$. Next, we combine the real and imaginary terms separately. The real terms are $64 + 24 = 88$, and the imaginary terms are $64i - 24i = 40i$. Therefore, the result of the multiplication is $88 + 40i$. This result will be crucial in the next step as we move towards simplifying the entire expression. Understanding this multiplication process is fundamental to working with complex numbers, and mastering it will enable you to tackle more complex problems with confidence. This meticulous approach ensures that we handle each term correctly, leading to an accurate result.

2. Subtraction Operation: $(8-3i) - (88+40i)$

Now that we've simplified the multiplication part of the expression, we move on to the subtraction. We have $(8-3i) - (88+40i)$. To perform this subtraction, we need to distribute the negative sign to both terms within the second set of parentheses. This means we change the sign of both the real and imaginary parts of the complex number $(88+40i)$. Distributing the negative sign, we get $8 - 3i - 88 - 40i$. Next, we combine like terms, which means grouping the real parts together and the imaginary parts together. The real parts are $8$ and $-88$, and their sum is $8 - 88 = -80$. The imaginary parts are $-3i$ and $-40i$, and their sum is $-3i - 40i = -43i$. Combining these results, we get the simplified complex number $-80 - 43i$. This is the final result of the expression. This step highlights the importance of careful sign manipulation when dealing with complex numbers, as a small error in sign can lead to a completely different answer. The process of combining like terms is a fundamental skill in algebra and is equally important when working with complex numbers. By meticulously following these steps, we ensure an accurate solution.

Identifying the Correct Answer

After performing the calculations, we have arrived at the simplified form of the expression: $-80 - 43i$. Now, let's compare this result with the given options to identify the correct answer. The options are:

A. $-96 - 37i$ B. $-96 + 37i$ C. $-80 - 43i$ D. $-80 + 43i

Comparing our result, $-80 - 43i$, with the options, we can clearly see that option C, $-80 - 43i$, matches our solution. Therefore, option C is the correct answer. This step is crucial in any mathematical problem-solving process, as it ensures that we not only arrive at a solution but also verify that our solution matches one of the given options. This careful comparison helps to minimize errors and ensures that we select the correct answer. In this case, the meticulous step-by-step approach we followed has led us to the correct solution, and we can confidently choose option C as the answer.

Conclusion

In this article, we embarked on a journey to solve the complex number expression $(8-3i)-(8-3i)(8+8i)$. Through a step-by-step approach, we first tackled the multiplication component, $(8-3i)(8+8i)$, using the distributive property (FOIL method). This yielded the result $88 + 40i$. Next, we performed the subtraction operation, distributing the negative sign and combining like terms, which led us to the simplified form $-80 - 43i$. By comparing our result with the given options, we confidently identified option C, $-80 - 43i$, as the correct answer. This exercise not only demonstrates the application of complex number arithmetic but also highlights the importance of a systematic and meticulous approach to problem-solving. Understanding the fundamental principles of complex number operations is crucial in various fields, and this detailed walkthrough serves as a valuable resource for anyone seeking to enhance their understanding of these fascinating numbers. The ability to confidently manipulate complex numbers opens doors to more advanced mathematical concepts and real-world applications. We hope this article has provided you with a clear and comprehensive understanding of how to solve such expressions, empowering you to tackle similar problems with ease and accuracy.