Simplify Radicals: Step-by-Step Solution

Hey guys! Let's dive into simplifying some radical expressions. Radicals might seem intimidating at first, but breaking them down step-by-step makes the whole process way more manageable. Today, we're going to tackle the expression -4√28 - 5√28 + 15√63 - 3√175. Buckle up, and let's get started!

Understanding the Basics of Simplifying Radicals

Before we jump into the problem, it's crucial to understand what simplifying radicals really means. At its core, simplifying radicals involves breaking down the number inside the square root (the radicand) into its prime factors. We’re looking for perfect square factors – numbers that have whole number square roots (like 4, 9, 16, 25, etc.). The goal is to pull out these perfect squares from under the radical sign, leaving us with a simplified expression. Why do we do this? Well, simplified radicals are easier to work with, compare, and understand. They also make it easier to see if we can combine terms, which is exactly what we'll be doing in our problem today. Think of it like reducing fractions – we're trying to express the radical in its most basic, understandable form. For example, √8 can be simplified because 8 has a perfect square factor of 4. We can rewrite √8 as √(4 * 2), which then simplifies to 2√2. This is the basic idea we'll apply to each term in our expression, but with slightly larger numbers and a few more steps. The key here is to methodically find those perfect square factors. Start with smaller perfect squares like 4 and 9, and work your way up. Sometimes, it helps to write out the prime factorization of the radicand to clearly see all the factors involved. This way, you can easily identify the pairs of identical factors that make up a perfect square. Remember, simplification is all about making things easier, so take your time and use the method that works best for you. With a little practice, you'll become a pro at simplifying radicals in no time!

Step 1: Simplifying -4√28 and -5√28

Okay, let's start with the first part of our expression: -4√28 - 5√28. Notice anything similar? Both terms have √28. This is a great starting point! To simplify √28, we need to find the largest perfect square that divides evenly into 28. Think about your perfect squares: 4, 9, 16, 25... Bingo! 4 is a perfect square (2 * 2 = 4), and it divides into 28 (28 ÷ 4 = 7). So, we can rewrite √28 as √(4 * 7). Now, here's the trick: remember that √(a * b) = √a * √b. So, √(4 * 7) becomes √4 * √7. We know √4 is 2, so we have 2√7. Fantastic! Now, let's plug this back into our original terms. -4√28 becomes -4 * (2√7) which equals -8√7. Similarly, -5√28 becomes -5 * (2√7) which equals -10√7. We've successfully simplified both terms! The next step is to combine these like terms. Since both terms now have the same radical part (√7), we can treat them like regular algebraic terms. Just add or subtract the coefficients (the numbers in front of the radical). In this case, we have -8√7 - 10√7. Think of it like -8