Simplifying Radicals: Multiplying √5(√14 - 8√5)

Hey guys! Today, we're diving into the fascinating world of radicals and tackling a problem that involves multiplying and simplifying them. It might seem a bit daunting at first, but trust me, with a step-by-step approach, it's totally manageable. We'll be working through the expression √5(√14 - 8√5), and by the end of this article, you'll be a pro at simplifying similar problems. So, grab your calculators (or your brains!), and let's get started!

Understanding the Basics of Radicals

Before we jump into the problem, let's quickly refresh our understanding of radicals. A radical is simply a root of a number. The most common radical is the square root (√), which asks, "What number, when multiplied by itself, equals the number under the root?" For example, √9 = 3 because 3 * 3 = 9. Radicals can also be cube roots, fourth roots, and so on, but for this problem, we're focusing on square roots.

When we're dealing with radicals, there are a few key rules to keep in mind. One important rule is that √(a * b) = √a * √b. This means we can break down the square root of a product into the product of the square roots. This will be super helpful when we're simplifying. Another crucial concept is simplifying radicals. We want to express the radical in its simplest form, which means the number under the root has no perfect square factors (other than 1). For example, √8 can be simplified to 2√2 because 8 = 4 * 2, and √4 = 2.

So, with these basics in mind, we're ready to tackle our main problem. Remember, the goal is to multiply √5(√14 - 8√5) and simplify the answer as much as possible. Let's break it down step by step to make it super clear and easy to follow. We'll be using the distributive property and the rules of radicals to get to our final simplified answer. Keep reading, and you'll see how it's done!

Step-by-Step Solution

Now, let's dive into the heart of the problem: simplifying the expression √5(√14 - 8√5). The first thing we need to do is apply the distributive property. This means we'll multiply √5 by each term inside the parentheses. So, we have:

√5 * √14 - √5 * 8√5

This might look a little intimidating, but don't worry, we'll take it one step at a time. Let's focus on the first term: √5 * √14. Remember the rule we talked about earlier, √(a * b) = √a * √b? We can use that here. √5 * √14 is the same as √(5 * 14), which equals √70. So, our first term simplifies to √70.

Now, let's tackle the second term: √5 * 8√5. Here, we're multiplying √5 by 8√5. We can rewrite this as 8 * √5 * √5. What's √5 * √5? It's simply 5! So, our second term becomes 8 * 5, which is 40. Now, remember the minus sign in front? So, this term is actually -40.

Putting it all together, we now have √70 - 40. But are we done yet? Not quite! We need to check if we can simplify √70 further. To do this, we look for perfect square factors of 70. The factors of 70 are 1, 2, 5, 7, 10, 14, 35, and 70. None of these (besides 1) are perfect squares, so √70 is already in its simplest form. Therefore, our final simplified answer is √70 - 40. See? Not so scary after all!

Breaking Down the Multiplication

Alright, let's really break down the multiplication process step by step to make sure we've got it nailed down. We started with the expression √5(√14 - 8√5). The key here is the distributive property, which tells us how to multiply a single term by a group of terms inside parentheses. Think of it like this: you're sharing the √5 with both terms inside the parentheses. So, you multiply √5 by √14 and then √5 by -8√5.

First up, we have √5 multiplied by √14. As we discussed earlier, when you multiply radicals with the same index (in this case, both are square roots), you can multiply the numbers inside the radicals. So, √5 * √14 becomes √(5 * 14), which is √70. Easy peasy!

Next, we have √5 multiplied by -8√5. This one might look a little trickier, but let's break it down. We can rewrite this as -8 * √5 * √5. Now, what happens when you multiply a square root by itself? You simply get the number inside the square root. So, √5 * √5 is just 5. That means our term becomes -8 * 5, which simplifies to -40.

So, after distributing and multiplying, we've got √70 - 40. This is a crucial step, and understanding how we got here is super important for tackling similar problems. Remember, the distributive property is your best friend when dealing with expressions like this. And don't forget the rule about multiplying radicals – it's a game-changer! Now that we've mastered the multiplication, let's move on to the final step: simplification.

Simplifying the Resulting Radical

Okay, we've successfully multiplied √5(√14 - 8√5) and arrived at the expression √70 - 40. Now, the final boss in this level is simplification. Simplifying radicals is all about making sure the number under the radical sign (the radicand) has no perfect square factors other than 1. In simpler terms, we want to see if we can pull out any perfect squares from under that square root.

So, let's focus on √70. To simplify it, we need to find the factors of 70. The factors are the numbers that divide evenly into 70. These are 1, 2, 5, 7, 10, 14, 35, and 70. Now, we need to check if any of these factors are perfect squares. A perfect square is a number that is the result of squaring a whole number (like 4, which is 2 squared, or 9, which is 3 squared). Looking at our list of factors, we see that none of them (other than 1) are perfect squares.

This means that √70 is already in its simplest form! We can't break it down any further. Sometimes, you'll find a perfect square factor, and you'll be able to simplify the radical. For example, if we had √72, we could simplify it to 6√2 because 72 has a perfect square factor of 36 (72 = 36 * 2, and √36 = 6). But in our case, √70 is as simple as it gets.

So, our simplified expression remains √70 - 40. This is our final answer! We've multiplied, distributed, and simplified, and we've conquered the radical problem. Give yourself a pat on the back – you've earned it!

Final Answer and Key Takeaways

Alright, let's recap. We started with the expression √5(√14 - 8√5) and, after a journey through multiplication and simplification, we arrived at our final answer: √70 - 40. Woohoo! But what did we learn along the way? Let's highlight some key takeaways to solidify our understanding.

First and foremost, we mastered the distributive property in the context of radicals. Remember, it's all about sharing that term outside the parentheses with each term inside. This is a fundamental skill in algebra, and you'll be using it all the time. Next, we reinforced our understanding of multiplying radicals. The rule √(a * b) = √a * √b is your friend! It allows you to combine radicals under one roof (or should we say, under one radical sign?).

We also honed our skills in simplifying radicals. This involves finding perfect square factors within the radicand and pulling them out. It's like giving those perfect squares a VIP pass out of the radical club! And finally, we learned to recognize when a radical is already in its simplest form. Sometimes, there are no perfect square factors to be found, and that's perfectly okay. Knowing when you're done is just as important as knowing how to start.

So, armed with these key takeaways, you're well-equipped to tackle similar radical problems. Remember, practice makes perfect, so don't be afraid to try more examples and challenge yourself. You've got this! And who knows, maybe you'll even start to enjoy simplifying radicals. Okay, maybe that's a stretch, but you'll definitely become more confident and proficient. Keep up the great work!