Simplifying Complex Numbers: A Step-by-Step Guide

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Hey math enthusiasts! Today, we're diving into the world of complex numbers and tackling a cool problem. The question is: Which expression is equivalent to (4+6i)2(4+6i)^2? Don't worry, it's not as scary as it looks. We'll break it down step by step, making sure you understand the 'why' behind each move. So, grab your calculators (or your brains!) and let's get started. We'll explore the problem, go over the core concepts, and finally arrive at the correct answer. This guide will walk you through the process, ensuring you not only find the solution but also grasp the underlying principles of complex number operations. Ready to conquer some math? Let's go!

Understanding Complex Numbers and the Problem

Alright, before we jump into the calculation, let's make sure we're all on the same page. 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. The 'a' part is called the real part, and the 'b' part (along with the 'i') is called the imaginary part. In our problem, we have the complex number (4 + 6i) and we need to square it. Squaring a complex number means multiplying it by itself. So, (4+6i)2(4 + 6i)^2 is the same as (4+6i)∗(4+6i)(4 + 6i) * (4 + 6i). Now, let's break down how to solve this, remembering the rules of algebra and how the imaginary unit, i, behaves.

Now, let's look at the given options:

A. 16−36i16 - 36i B. 20+48i20 + 48i C. −20+48i-20 + 48i

Our task is to find which of these options is equal to (4+6i)2(4+6i)^2. To do this, we need to carefully perform the multiplication and simplify the expression. We will be using the distributive property (also known as the FOIL method, which stands for First, Outer, Inner, Last) to multiply the complex numbers. Then, we will use the property of the imaginary unit ii (i.e. i2=−1i^2 = -1) to simplify the expression further. We'll do it step by step, and I promise, it's easier than it sounds! Remember, practice makes perfect. The more you do these kinds of problems, the more comfortable you'll become with complex numbers. So, buckle up; we're about to make this complex number problem a breeze. Let's start with the basics.

Core Concepts and the Imaginary Unit

Before we dive into the calculations, let's quickly review the core concepts. The imaginary unit, denoted by i, is defined as the square root of -1. This is a crucial concept because it allows us to work with the square roots of negative numbers. A fundamental property of i is that i squared (i²), is equal to -1. This is where things get interesting and distinct from regular algebra. This property will be key in simplifying our expression. When we multiply complex numbers, we need to remember this property. Every time we see i², we replace it with -1. This simple substitution is what turns complex number calculations into a manageable process. Keep this in mind, and you'll be well-equipped to solve the given problem and others like it. This understanding is the cornerstone of all complex number operations. Now, with these concepts in mind, let's move forward and apply them to solve the problem at hand. We're getting closer to that solution, I can feel it!

Step-by-Step Solution

Alright, time to get our hands dirty and solve this problem step by step. We have (4+6i)2(4 + 6i)^2, which is the same as (4+6i)∗(4+6i)(4 + 6i) * (4 + 6i). Let's use the distributive property (FOIL method) to expand this:

  • First: Multiply the first terms: 4 * 4 = 16
  • Outer: Multiply the outer terms: 4 * (6i) = 24i
  • Inner: Multiply the inner terms: (6i) * 4 = 24i
  • Last: Multiply the last terms: (6i) * (6i) = 36i²

Now, let's put it all together: 16 + 24i + 24i + 36i². Remember that i² = -1. So, we can replace 36i² with 36 * (-1) = -36. This simplifies our expression to 16 + 24i + 24i - 36. Combine like terms (the real parts and the imaginary parts): (16 - 36) + (24i + 24i) = -20 + 48i. Voila! We have our simplified expression. Easy peasy, right? Now, let's go over the options again and match the answer.

Now, let's write out each step in detail so that it's super clear:

  1. Expand the square: (4+6i)2=(4+6i)∗(4+6i)(4 + 6i)^2 = (4 + 6i) * (4 + 6i). This step means we are simply rewriting the expression to show the multiplication explicitly.
  2. Apply the distributive property (FOIL):
    • (4∗4)+(4∗6i)+(6i∗4)+(6i∗6i)(4 * 4) + (4 * 6i) + (6i * 4) + (6i * 6i)
    • =16+24i+24i+36i2= 16 + 24i + 24i + 36i^2 This step is all about multiplying out the terms.
  3. Simplify i²: Recall that i2=−1i^2 = -1. Substitute this into our expression:
    • 16+24i+24i+36(−1)16 + 24i + 24i + 36(-1)
    • =16+24i+24i−36= 16 + 24i + 24i - 36
  4. Combine like terms: Combine the real parts (16 and -36) and the imaginary parts (24i and 24i):
    • (16−36)+(24i+24i)(16 - 36) + (24i + 24i)
    • =−20+48i= -20 + 48i

This methodical breakdown ensures we've handled all the steps carefully and correctly. The key here is not to rush. The slower and more detailed you are, the less likely you are to make a mistake. Congrats, we got this!

The Correct Answer and Why

So, after all that hard work, which option is the correct one? Drumroll, please... C. -20 + 48i is the correct answer! Our step-by-step solution led us directly to this expression. We expanded the square, applied the distributive property, remembered that i² = -1, and then combined the real and imaginary parts. This methodical process ensured that we arrived at the correct answer. The other options, A and B, are incorrect because they do not match the result of our calculations. Option A (16 - 36i) and Option B (20 + 48i) are not the same as the simplified form of (4+6i)2(4 + 6i)^2, which is −20+48i-20 + 48i. By carefully following each step and understanding the properties of complex numbers, we can confidently eliminate these incorrect options and arrive at the solution. Awesome work, everyone! You've successfully solved the problem and gained a deeper understanding of complex numbers.

Now, let's break down why the other options are wrong:

  • A. 16−36i16 - 36i: This option is incorrect because it doesn't align with the correct expansion and simplification. It seems like a result of some miscalculation or mistake during the multiplication and combining of terms. This result doesn't follow the proper rules of complex number arithmetic.
  • B. 20+48i20 + 48i: This option is also incorrect. There could be errors in the distributive property application or incorrect handling of the imaginary unit (i2i^2). Specifically, it suggests mistakes in the multiplication of either the real or imaginary parts during the initial expansion.

By comparing these incorrect options to our correct answer, you can see how important it is to be precise in each step. Little mistakes can lead to entirely different results, highlighting the need for careful calculations and a strong understanding of complex number properties.

Tips and Tricks for Solving Similar Problems

Want to become a complex number whiz? Here are a few tips and tricks that will help you solve similar problems with ease:

  • Master the FOIL Method: The distributive property (or FOIL method) is your best friend when multiplying complex numbers. Make sure you're comfortable with it. Practice, practice, practice!
  • Remember i² = -1: This is the most important property to remember. It's the key to simplifying complex number expressions. Write it down, highlight it, tattoo it on your brain – whatever works!
  • Combine Like Terms: Always combine the real and imaginary parts separately. This makes your final answer clean and easy to read. This helps in minimizing calculation errors and organizing your work.
  • Double-Check Your Work: It's always a good idea to go back and check your calculations. A small mistake can lead to a wrong answer. Take your time; the goal is accuracy.
  • Practice Regularly: The more you practice, the better you'll get. Try different problems with different complex numbers to build your confidence. Look for a variety of questions to give yourself a full understanding.
  • Use the Right Tools: A calculator with complex number functions can be very helpful. But don't rely on it completely. Make sure you understand the steps involved.
  • Simplify as You Go: Simplify each step. This keeps the numbers manageable and reduces the chances of errors.

By following these tips, you'll be well on your way to mastering complex numbers. Remember, practice and consistency are the keys to success. Keep at it, and you'll find these problems becoming easier and more enjoyable. These are not just tips for complex numbers; they're valuable strategies for any kind of math problem.

Conclusion: You Got This!

Alright, folks, we've reached the end of our journey! We started with a complex number problem, broke it down step by step, and found the correct answer. We've gone over the core concepts, practiced our calculations, and learned some valuable tips and tricks. Remember, complex numbers might seem tricky at first, but with practice and a good understanding of the basics, you can conquer any problem. So keep practicing, keep learning, and don't be afraid to challenge yourself. You've got this! And remember, math is a journey, not a destination. Embrace the process, and enjoy the ride. Keep up the awesome work, and I'll see you in the next lesson!

If you have any questions or want to try another problem, feel free to ask. Happy calculating, and keep those math muscles flexing! You've officially leveled up your complex number skills. Great job today, everyone!