Simplify Radicals: A Step-by-Step Guide
Hey guys! Today, we're diving into the world of simplifying radical expressions. Radicals might seem intimidating at first, but trust me, once you break them down, they're not so scary. We're going to tackle the expression $5 \sqrt{125}-4 \sqrt{8}-3 \sqrt{45}$ step-by-step, so you can master these types of problems. Simplifying radicals involves expressing them in their simplest form, which means removing any perfect square factors from under the radical sign. This not only makes the expression look cleaner but also makes it easier to combine like terms.
Understanding the Basics of Simplifying Radicals
Before we jump into the main problem, let's quickly review the basics of simplifying radicals. A radical expression consists of a radical symbol (√), a radicand (the number under the radical symbol), and an index (which indicates the root to be taken; if no index is written, it's assumed to be 2, meaning square root). Simplifying radicals primarily involves identifying and extracting perfect square factors from the radicand. For example, to simplify √20, we recognize that 20 can be factored into 4 × 5, where 4 is a perfect square (2²). Thus, √20 can be written as √(4 × 5) = √4 × √5 = 2√5. The key is to find the largest perfect square factor to make the simplification process efficient. Remember, simplifying radicals is not just about getting the right answer, it's about understanding the underlying principles of numbers and their factors. By mastering the basics, you'll be able to tackle more complex radical expressions with confidence. This foundation will also be crucial as you progress in mathematics, where radicals appear in various contexts, such as solving quadratic equations and dealing with trigonometric functions. So, let's solidify this understanding and get ready to simplify some radicals!
Breaking Down $5 \sqrt{125}$
Let's start with the first term: $5 \sqrt125}$. Our mission here is to simplify radicals, specifically the square root of 125. To do this, we need to find the largest perfect square that divides evenly into 125. Think of perfect squares like 4, 9, 16, 25, and so on. Can you see which one fits? Bingo! 25 is a perfect square (5 * 5 = 25), and it divides into 125 five times (125 = 25 * 5). Now we can rewrite our expression = 5 \sqrt25 * 5}$. The beauty of simplifying radicals is that we can split the square root of a product into the product of square roots = 5 * \sqrt25} * \sqrt{5}$. We know the square root of 25 is 5, so we have * \sqrt5} = 5 * 5 * \sqrt{5}$. Finally, multiply those outside numbers together = 25 \sqrt{5}$. See? Not too bad! We've successfully simplified radicals in the first term. The secret is to always look for those perfect square factors. This step-by-step process not only simplifies the expression but also builds a solid understanding of how radicals work. Remember, practice makes perfect, so the more you break down radicals like this, the easier it will become. Let's move on to the next term and continue our journey of simplifying radicals!
Simplifying $-4 \sqrt{8}$
Now, let's tackle the second term: $-4 \sqrt8}$. We're on a roll with simplifying radicals, so let's keep that momentum going! Just like before, we need to find the largest perfect square that divides evenly into 8. What do you think? Yep, it's 4 (2 * 2 = 4), and 8 is 4 * 2. So we can rewrite our term as = -4 \sqrt4 * 2}$. Time to use our handy rule of splitting the square root = -4 * \sqrt4} * \sqrt{2}$. We know that the square root of 4 is 2, so we have * \sqrt2} = -4 * 2 * \sqrt{2}$. Now, multiply those outside numbers together = -8 \sqrt{2}$. Fantastic! We've successfully simplified radicals in the second term. It's all about spotting those perfect square factors and breaking things down step by step. As you can see, simplifying radicals becomes a lot less daunting when you take it one step at a time. Each term is like a mini-puzzle, and the satisfaction of finding those perfect square factors is pretty awesome. So, let's keep this winning streak going and move on to the final radical term. We're becoming simplifying radicals pros at this point!
Tackling $-3 \sqrt{45}$
Alright, guys, we're on the final stretch! Let's conquer the last term: $-3 \sqrt45}$. We've become quite the experts at simplifying radicals by now, so let's put those skills to work. Time to find the largest perfect square that divides evenly into 45. Think about it... which number fits the bill? You got it! 9 is a perfect square (3 * 3 = 9), and 45 is 9 * 5. So, we rewrite our term as = -3 \sqrt9 * 5}$. Let's split that square root like we've done before = -3 * \sqrt9} * \sqrt{5}$. The square root of 9 is 3, so we have * \sqrt5} = -3 * 3 * \sqrt{5}$. Multiply the outside numbers together = -9 \sqrt{5}$. Boom! We've successfully simplified radicals in the final term. We're on fire! This term was simplified in the same way, which makes it easier to tackle. Remember, simplifying radicals is like building blocks – each term you simplify adds to your understanding and makes the next one even easier. Now that we've simplified all the individual terms, it's time to put them all together and see what we get. Let's wrap this up and show off our simplifying radicals mastery!
Combining Like Terms
Okay, now for the grand finale! We've simplified radicals in each term, so let's recap what we have: $25 \sqrt5} - 8 \sqrt{2} - 9 \sqrt{5}$. Now, we need to combine like terms. Remember, like terms are those that have the same radical part. In our expression, we have two terms with $\sqrt{5}$$ and $-9 \sqrt5}$. The term $-8 \sqrt{2}$ is different because it has a $\sqrt{2}$, so we can't combine it with the others. To combine like terms, we simply add or subtract their coefficients (the numbers in front of the radical). So, let's focus on those $\sqrt{5}$ terms - 9 \sqrt5} = (25 - 9) \sqrt{5} = 16 \sqrt{5}$. Now, let's bring back the term with $\sqrt{2}$, which we couldn't combine - 8 \sqrt{2}$. And that's it! We've combined all the like terms. Simplifying radicals and combining like terms is a powerful skill, and you've just mastered it. Remember, the key is to break down each term, find those perfect square factors, and then combine the terms that are similar. You've done an awesome job following along, and you should be super proud of your simplifying radicals skills. Now, go out there and conquer more radical expressions!
Final Answer
So, after simplifying radicals and combining like terms, our final answer is: $16 \sqrt{5} - 8 \sqrt{2}$. You did it! By breaking down each step and focusing on simplifying radicals, we've successfully solved this problem. Remember, mathematics is all about practice and understanding the underlying concepts. We hope this step-by-step guide has helped you grasp the art of simplifying radicals and feel more confident in tackling similar problems. Keep practicing, keep exploring, and you'll become a math whiz in no time!
