Multiplying Complex Numbers A Step By Step Guide

Hey guys! Today, we're diving into the world of complex numbers and tackling a problem that might seem a bit intimidating at first glance: multiplying $(3+\sqrt{-16})(6-\sqrt{-64})$. Don't worry, we'll break it down step by step so it becomes super clear. Think of this as your friendly guide to navigating the complex plane – no prior experience required!

Understanding the Basics: Imaginary Numbers and Complex Numbers

Before we jump into the multiplication, let's quickly refresh our understanding of imaginary and complex numbers. Remember that the square root of a negative number isn't a real number. That's where imaginary numbers come in. The imaginary unit, denoted by i, is defined as the square root of -1 (i.e., $i = \sqrt{-1}$). This little guy is the key to unlocking the complex number world.

Now, a complex number is simply 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 a part is called the real part, and the bi part is called the imaginary part. So, for example, 3 + 2i is a complex number, where 3 is the real part and 2i is the imaginary part.

Think of complex numbers as an extension of the real number line. While real numbers live on a single line, complex numbers live on a plane, often called the complex plane. The horizontal axis represents the real part, and the vertical axis represents the imaginary part. This visual representation can be super helpful when we're working with complex numbers.

In our problem, we have two complex numbers: $(3+\sqrt{-16})$ and $(6-\sqrt{-64})$. Notice the square roots of negative numbers? That's our cue to bring in the imaginary unit i. Let's simplify these numbers first before we multiply them. This is a crucial step, guys, because it makes the multiplication process much smoother. By understanding these fundamental concepts, we can confidently approach the problem at hand and demystify complex number multiplication.

Step 1: Simplifying the Square Roots of Negative Numbers

The first step in tackling this problem is simplifying the square roots of the negative numbers. Remember, $\sqrt{-1} = i$. So, we can rewrite $\sqrt{-16}$ and $\sqrt{-64}$ using i. This is where the magic happens, guys! This is where we transform the seemingly complicated into something manageable.

Let's start with $\sqrt{-16}$. We can rewrite this as $\sqrt{16 \cdot -1}$. Using the property of square roots that $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$, we can further break this down into $\sqrt{16} \cdot \sqrt{-1}$. We know that $\sqrt{16} = 4$ and $\sqrt{-1} = i$, so $\sqrt{-16} = 4i$. See how we transformed a potentially confusing term into a simple imaginary number? This is the power of understanding the basics!

Now, let's do the same for $\sqrt{-64}$. We can rewrite this as $\sqrt{64 \cdot -1}$, which is equal to $\sqrt{64} \cdot \sqrt{-1}$. We know that $\sqrt{64} = 8$ and $\sqrt{-1} = i$, so $\sqrt{-64} = 8i$. Boom! Another simplification under our belts. We're making great progress, guys.

By simplifying these square roots, we've transformed our original complex numbers into a much more manageable form. Now we have $(3 + 4i)$ and $(6 - 8i)$. This makes the multiplication process significantly easier. Remember, simplifying first is often the key to unlocking complex problems. This step is not just about calculation; it's about strategic problem-solving. We are setting ourselves up for success in the next step.

Step 2: Rewriting the Expression with Simplified Terms

Now that we've simplified the square roots, let's rewrite our original expression with the simplified terms. This will make the multiplication process much clearer and less prone to errors. It's like decluttering your workspace before starting a project – it helps you focus and work more efficiently.

Our original expression was $(3+\sqrt{-16})(6-\sqrt{-64})$. We found that $\sqrt{-16} = 4i$ and $\sqrt{-64} = 8i$. So, we can substitute these values into our expression. This gives us $(3 + 4i)(6 - 8i)$. See how much cleaner that looks? We've replaced the potentially confusing square roots with simple imaginary terms. This is a crucial step in making the problem more approachable.

Rewriting the expression is not just about substituting values; it's about transforming the problem into a form that we can easily work with. We've essentially translated the problem from one language (with square roots of negative numbers) to another language (with imaginary units). This translation is a key skill in mathematics, allowing us to apply familiar techniques to new situations. By taking this step, we've set ourselves up for the next stage: multiplying these complex numbers.

Step 3: Multiplying the Complex Numbers Using the Distributive Property (FOIL)

Alright, guys, now comes the fun part: actually multiplying the complex numbers! We'll use the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last), to make sure we multiply each term correctly. Think of FOIL as a roadmap for navigating the multiplication process. It helps us keep track of which terms we've multiplied and which we still need to address. This is where the magic really happens, where we combine the real and imaginary parts to get our final answer.

Our expression is $(3 + 4i)(6 - 8i)$. Let's break it down using FOIL:

• First: Multiply the first terms in each parenthesis: 3 * 6 = 18
• Outer: Multiply the outer terms: 3 * (-8i) = -24i
• Inner: Multiply the inner terms: 4i * 6 = 24i
• Last: Multiply the last terms: 4i * (-8i) = -32i²

So, after applying FOIL, we have 18 - 24i + 24i - 32i². Notice anything interesting? The -24i and +24i terms cancel each other out! This often happens when multiplying complex conjugates (more on that later). But we're not done yet. We still have that -32i² term to deal with. Remember, i is the square root of -1, so i² is equal to -1. This is a crucial piece of information, guys, because it allows us to simplify the expression further.

By carefully applying the distributive property and remembering the definition of i², we're making steady progress towards our solution. This step highlights the importance of both procedural knowledge (how to multiply) and conceptual understanding (what i² means). We're not just crunching numbers; we're understanding the underlying principles.

Step 4: Simplifying the Expression Using $i^2 = -1$

Okay, guys, let's talk about the heart of simplifying complex number expressions: the golden rule $i^2 = -1$. This little equation is the key to unlocking a lot of simplification potential. It allows us to transform imaginary terms into real terms, which is essential for expressing our final answer in the standard form of a complex number (a + bi).

In the previous step, we arrived at the expression 18 - 24i + 24i - 32i². We noticed that the -24i and +24i terms canceled each other out, leaving us with 18 - 32i². Now, let's focus on that -32i² term. Remember, i² = -1, so we can substitute -1 for i² in our expression. This gives us 18 - 32(-1).

Now we have a simple arithmetic problem to solve. -32 multiplied by -1 is 32. So, our expression becomes 18 + 32. See how we transformed an imaginary term into a real term using the magic of i² = -1? This is a fundamental technique in complex number manipulation. By understanding this rule, we can confidently navigate the world of complex numbers and simplify even the most intimidating expressions.

Step 5: Combining Like Terms to Get the Final Answer

We're almost there, guys! We've done the hard work of simplifying the square roots, multiplying the terms, and using the rule i² = -1. Now, all that's left is to combine any remaining like terms to arrive at our final answer. This is like the final polish on a beautifully crafted piece of work – it brings everything together and presents the result in its clearest form.

In the previous step, we had the expression 18 + 32. These are both real numbers, so we can simply add them together. 18 + 32 = 50. And that's it! We've successfully multiplied the complex numbers and simplified the result. Our final answer is 50.

Notice that the imaginary terms completely canceled out in this problem. This doesn't always happen, but it's a nice surprise when it does! It highlights the beauty and elegance of complex number arithmetic. Our final answer, 50, is a real number. This means that the product of the two complex numbers $(3+\sqrt{-16})$ and $(6-\sqrt{-64})$ lies on the real number line in the complex plane. This is a great example of how complex number multiplication can sometimes lead to surprising results. By combining like terms, we've completed the final step in our journey and arrived at our solution. Congratulations, guys! We've conquered complex number multiplication!

Final Answer

So, guys, after all that hard work, we've arrived at our final answer! The product of $(3+\sqrt{-16})(6-\sqrt{-64})$ is 50. Not so scary after all, right? We broke down the problem step by step, from simplifying the square roots to using the distributive property and the crucial rule i² = -1. This systematic approach is key to tackling any math problem, especially those involving complex numbers.

Remember, guys, the key to mastering complex numbers is understanding the basics and practicing consistently. Don't be afraid to make mistakes – they're a part of the learning process. The more you work with these concepts, the more comfortable you'll become. And who knows, maybe you'll even start to find complex numbers... well, complexly fascinating! Keep up the great work, and I'll see you in the next math adventure!