Solving 3√20 + √72 - 2√5 A Step-by-Step Guide
Hey guys! Ever stumbled upon a math problem that looks like it belongs in a superhero movie? Well, let's face it, square roots can seem intimidating at first glance. But don't worry, we're here to break down one of those seemingly complex problems into bite-sized, easily digestible steps. Today, we're tackling the expression 3√20 + √72 - 2√5. Sounds like a mouthful, right? But trust me, by the end of this guide, you'll be simplifying expressions like a math whiz! We're not just going to give you the answer; we're going to walk you through the entire process, so you understand the why behind each step. This isn't just about getting the right answer; it's about building your math skills and confidence. So, grab your pencils, notebooks, and maybe a cup of coffee, because we're about to dive deep into the world of square roots. We'll start by understanding the fundamental properties of square roots, then move on to simplifying individual terms, and finally, we'll combine everything to get to our final answer. Think of this as a mathematical adventure, where each step brings us closer to the treasure – the simplified solution! And remember, math isn't about memorizing formulas; it's about understanding the logic and reasoning behind them. So, let's get started and unravel the mystery of 3√20 + √72 - 2√5 together! This journey will not only help you solve this specific problem but also equip you with the tools to tackle similar problems with ease and confidence. Remember, practice makes perfect, and understanding the core concepts is key to mastering mathematics. So, let's embark on this mathematical quest and conquer those square roots!
Breaking Down the Basics of Square Roots
Okay, before we jump into the main problem, let's rewind a bit and make sure we're all on the same page when it comes to square roots. Think of a square root as the opposite of squaring a number. For instance, the square of 5 (5²) is 25, so the square root of 25 (√25) is 5. Easy peasy, right? But things get a little more interesting when we deal with numbers that aren't perfect squares, like 20 or 72. That's where the concept of simplifying square roots comes in. The key idea here is to find the largest perfect square that divides evenly into the number under the square root. For example, 20 can be written as 4 * 5, where 4 is a perfect square (2 * 2). This allows us to rewrite √20 as √(4 * 5). Now, here's a crucial property of square roots: the square root of a product is the product of the square roots. In mathematical terms, √(a * b) = √a * √b. Applying this to our example, we get √(4 * 5) = √4 * √5. And since we know √4 is 2, we can simplify √20 to 2√5. See how we took a seemingly complicated square root and broke it down into a simpler form? This technique is going to be our best friend as we tackle the original problem. Understanding this fundamental principle is crucial because it allows us to manipulate and simplify expressions involving square roots. Without this understanding, dealing with expressions like 3√20 + √72 - 2√5 would be significantly more challenging. So, make sure you've got this concept down pat before moving on to the next step. It's like building a house – you need a strong foundation before you can start adding the walls and roof. And in the world of mathematics, a strong understanding of the basics is the foundation for tackling more complex problems. So, let's keep this principle in mind as we move forward and conquer the world of square roots!
Simplifying Individual Terms: 3√20, √72, and 2√5
Now that we've brushed up on our square root basics, let's get our hands dirty and simplify each term in the expression 3√20 + √72 - 2√5 individually. This is like taking apart a machine to see how each piece works before putting it back together. Let's start with the first term: 3√20. Remember how we talked about finding perfect square factors? We can rewrite 20 as 4 * 5, where 4 is a perfect square. So, 3√20 becomes 3√(4 * 5). Using the property √(a * b) = √a * √b, we can rewrite this as 3 * √4 * √5. We know √4 is 2, so we have 3 * 2 * √5, which simplifies to 6√5. Voila! We've simplified the first term. Next up, we have √72. This one might seem a bit trickier, but let's break it down. What's the largest perfect square that divides into 72? If you're thinking 36, you're spot on! 72 can be written as 36 * 2. So, √72 becomes √(36 * 2), which is √36 * √2. Since √36 is 6, we can simplify √72 to 6√2. Awesome! We're two terms down, one to go. Now, let's look at the last term: 2√5. Notice anything special about this term? It's already in its simplest form! The number under the square root, 5, doesn't have any perfect square factors (other than 1, of course). So, we can leave 2√5 as it is. Simplifying each term individually is like preparing the ingredients before you start cooking. It makes the final assembly much easier and smoother. By breaking down each term into its simplest form, we're setting ourselves up for success in the next step, where we'll combine everything to get the final answer. Remember, patience and attention to detail are key in mathematics. Taking the time to simplify each term correctly will prevent errors and ensure that we arrive at the correct solution. So, let's keep up the great work and move on to the final stage of our mathematical journey!
Combining Like Terms: Putting It All Together
Alright, guys, we've done the prep work, and now it's time for the grand finale! We've simplified each term in the expression 3√20 + √72 - 2√5, and we're ready to put it all together. This is like the assembly line in a factory, where all the individual components come together to form the final product. So, let's recap what we've got: We simplified 3√20 to 6√5. We simplified √72 to 6√2. And 2√5 was already in its simplest form. Now, we can rewrite the original expression as 6√5 + 6√2 - 2√5. Notice anything interesting? We have two terms with √5: 6√5 and -2√5. These are called like terms, and just like in algebra, we can combine them. Think of √5 as a variable, like 'x'. So, 6√5 - 2√5 is like saying 6x - 2x, which is 4x. In our case, it's 4√5. So, now we have 4√5 + 6√2. Can we simplify this any further? Nope! √5 and √2 are different, so these terms cannot be combined. It's like trying to add apples and oranges – they're just not the same thing. Therefore, our final simplified expression is 4√5 + 6√2. Ta-da! We've done it! We took a seemingly complex expression and simplified it step-by-step, using our knowledge of square roots and some basic algebra. Combining like terms is a crucial skill in mathematics, and it's something you'll use again and again. It's like having a universal key that unlocks many mathematical doors. By recognizing and combining like terms, you can simplify expressions, solve equations, and even tackle more advanced mathematical concepts. So, pat yourselves on the back, guys! You've successfully navigated the world of square roots and emerged victorious. Remember, math is a journey, not a destination. The more you practice and understand the underlying concepts, the more confident and skilled you'll become. So, keep exploring, keep questioning, and keep simplifying!
Final Answer and Key Takeaways
Let's wrap things up by stating our final answer and highlighting some key takeaways from our mathematical adventure. After simplifying the expression 3√20 + √72 - 2√5, we arrived at the answer: 4√5 + 6√2. Now, let's reflect on the journey we took to get here. We started by understanding the basics of square roots and how to simplify them by finding perfect square factors. This is like learning the alphabet before you can read a book – it's a fundamental skill that unlocks more complex concepts. Then, we tackled each term in the expression individually, simplifying 3√20 to 6√5 and √72 to 6√2. This step-by-step approach is crucial in problem-solving. It's like breaking down a large task into smaller, manageable steps. Finally, we combined like terms, recognizing that 6√5 and -2√5 could be combined to give us 4√5. This highlights the importance of recognizing patterns and applying algebraic principles to simplify expressions. So, what are the key takeaways from this exercise? First, understanding the properties of square roots is essential. Knowing that √(a * b) = √a * √b is a powerful tool for simplification. Second, identifying perfect square factors is crucial. This allows you to break down square roots into simpler forms. Third, combining like terms is a fundamental algebraic skill that simplifies expressions and makes them easier to work with. And perhaps most importantly, a step-by-step approach is key to solving complex problems. By breaking down the problem into smaller parts, we made it much easier to manage and solve. Remember, guys, math is not just about getting the right answer; it's about the process of getting there. It's about developing problem-solving skills, logical thinking, and a deeper understanding of the world around us. So, keep practicing, keep exploring, and keep challenging yourselves. And the next time you encounter a seemingly complex math problem, remember the journey we took today and break it down step-by-step. You've got this!
