Simplifying Complex Numbers Step-by-Step Solution For (-2i^4)(3i^5)(i^3)

Hey there, math enthusiasts! Today, we're diving into the fascinating world of complex numbers and tackling a problem that might seem a bit daunting at first glance. But fear not! We're going to break it down step by step, making it super easy to understand. Our mission: to simplify the expression $(-2i^4)(3i^5)(i^3)$ and find the correct answer from the given choices. Let's get started!

Understanding Complex Numbers: The Foundation of Our Journey

Before we jump into the calculation, let's take a moment to understand the basics of complex numbers. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit. The imaginary unit, i, is defined as the square root of -1, which means $i^2 = -1$. This seemingly simple definition unlocks a whole new realm of mathematical possibilities.

The beauty of complex numbers lies in their ability to represent solutions to equations that have no real number solutions. For instance, the equation $x^2 + 1 = 0$ has no solutions within the set of real numbers, but it has two solutions in the complex number system: i and -i. Complex numbers are not just abstract mathematical concepts; they have practical applications in various fields, including electrical engineering, quantum mechanics, and signal processing. They help us model and understand phenomena that cannot be described using real numbers alone.

Now, let's delve deeper into the powers of i. We know that $i^2 = -1$. So, what about $i^3$? We can express $i^3$ as $i^2 * i = -1 * i = -i$. Similarly, $i^4 = i^2 * i^2 = (-1) * (-1) = 1$. This pattern is crucial for simplifying expressions involving complex numbers. Notice how the powers of i cycle through four values: i, -1, -i, and 1. This cyclical nature is what makes complex number calculations so elegant and manageable.

Powers of i: A Quick Reference

To make things even clearer, let's summarize the powers of i:

• $i^1 = i$
• $i^2 = -1$
• $i^3 = -i$
• $i^4 = 1$

This pattern repeats for higher powers of i. For example, $i^5 = i^4 * i = 1 * i = i$, and so on. Keeping this cycle in mind will help you simplify complex number expressions quickly and accurately. So, with a solid grasp of what complex numbers are and how the powers of i work, we're now well-equipped to tackle our main problem. Let's move on to simplifying the given expression!

Step-by-Step Simplification: Cracking the Code of (-2i⁴⁾⁽³ⁱ5)(i^3)

Alright, guys, let's get our hands dirty and simplify the expression $(-2i^4)(3i^5)(i^3)$. The key here is to break it down into smaller, manageable parts and use the properties of i we discussed earlier. We'll go through each term, simplify it, and then combine them to get our final answer.

First, let's look at the term $-2i^4$. We know that $i^4 = 1$, so $-2i^4$ becomes $-2 * 1 = -2$. That's the first part sorted out! Now, let's move on to the next term, $3i^5$. We can rewrite $i^5$ as $i^4 * i$. Since $i^4 = 1$, $i^5$ simplifies to $1 * i = i$. So, $3i^5$ becomes $3 * i = 3i$. We're making great progress, aren't we?

Finally, let's tackle the last term, $i^3$. As we discussed earlier, $i^3 = -i$. Now that we've simplified each term individually, let's put them all together. Our expression $(-2i^4)(3i^5)(i^3)$ now looks like this: $(-2)(3i)(-i)$. We're almost there! Now, it's just a matter of multiplying these terms together.

Let's start by multiplying the numbers: $-2 * 3 = -6$. So, we have $-6 * i * (-i)$. Now, let's multiply the i terms: $i * (-i) = -i^2$. But remember, $i^2 = -1$, so $-i^2$ becomes $-(-1) = 1$. This is a crucial step, so make sure you're following along. Now, we have $-6 * 1$, which equals -6. And there you have it! We've successfully simplified the expression.

Breaking Down the Calculation:

To recap, here's the step-by-step breakdown:

1. $-2i^4 = -2 * 1 = -2$
2. $3i^5 = 3 * i^4 * i = 3 * 1 * i = 3i$
3. $i^3 = -i$
4. $(-2)(3i)(-i) = -6 * i * (-i) = -6 * (-i^2)$
5. $-6 * (-(-1)) = -6 * 1 = -6$

So, the simplified form of $(-2i^4)(3i^5)(i^3)$ is -6. Now, let's compare our answer with the given choices and see which one matches.

Matching the Answer: Which Choice is the Winner?

We've done the hard work and simplified the expression to -6. Now, let's look at the answer choices provided:

A. -6i B. -6 C. 1 D. 6 E. 6i

Comparing our result with the choices, we can see that option B. -6 perfectly matches our answer. Woohoo! We've successfully solved the problem. This whole process demonstrates the power of breaking down a complex problem into smaller, manageable steps. It's a technique that's useful not just in math, but in all aspects of problem-solving.

Why Other Options Are Incorrect

Just for the sake of completeness, let's briefly discuss why the other options are incorrect:

• A. -6i: This answer has an imaginary component, but our simplified expression is a real number (-6).
• C. 1: This is a positive real number, but our result is a negative real number.
• D. 6: This is the positive counterpart of our correct answer. A common mistake might be missing the negative sign during the simplification process.
• E. 6i: Again, this answer has an imaginary component, while our result is a real number.

Understanding why incorrect options are wrong is just as important as knowing why the correct option is right. It helps solidify your understanding of the concepts involved.

Key Takeaways: Mastering Complex Number Simplification

Okay, guys, we've reached the end of our journey through this complex number problem. Before we wrap up, let's recap the key takeaways. These are the concepts and techniques that will help you tackle similar problems with confidence.

First and foremost, remember the definition of the imaginary unit: $i = \sqrt{-1}$, and consequently, $i^2 = -1$. This is the foundation upon which all complex number calculations are built. Next, understand the cyclical nature of the powers of i: i, -1, -i, 1. Being able to quickly recall these values will save you time and prevent errors.

When simplifying complex expressions, break them down into smaller parts. Simplify each term individually and then combine them. This approach makes the problem much more manageable. Also, pay close attention to signs, especially when dealing with negative numbers and the imaginary unit. A small mistake in a sign can lead to a completely different answer.

Tips for Success with Complex Numbers:

• Memorize the powers of i: Know that $i^1 = i$, $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$.
• Break down expressions: Simplify each term separately before combining them.
• Watch out for signs: Be extra careful with negative signs and the imaginary unit.
• Practice, practice, practice: The more you work with complex numbers, the more comfortable you'll become.

Complex numbers might seem intimidating at first, but with a solid understanding of the basics and a step-by-step approach, you can conquer any problem they throw your way. So, keep practicing, keep exploring, and keep having fun with math! You've got this!

Conclusion: Celebrating Our Complex Number Victory

We did it, guys! We successfully simplified the expression $(-2i^4)(3i^5)(i^3)$ and found the correct answer: -6. This wasn't just about getting the right answer; it was about understanding the underlying concepts and developing a systematic approach to problem-solving. We learned about the definition of complex numbers, the powers of i, and how to simplify complex expressions step by step.

Complex numbers are a fascinating and powerful tool in mathematics. They extend our ability to solve equations and model real-world phenomena. So, don't shy away from them! Embrace the challenge, and you'll be amazed at what you can achieve. Remember, math is not just about numbers and formulas; it's about critical thinking, problem-solving, and the joy of discovery.

I hope this step-by-step guide has been helpful and insightful. Keep practicing, and you'll become a complex number whiz in no time! Until next time, happy calculating!