Simplifying Radicals A Step-by-Step Guide To -2/3 Sqrt(63x^3)

Understanding the Problem

Okay guys, let's dive into simplifying this radical expression: $-\frac{2}{3} \sqrt{63 x^3}$. At first glance, it might look a little intimidating, but don't worry, we'll break it down step by step. The core idea behind simplifying radicals is to identify perfect square factors within the radicand (the expression under the square root) and pull them out. Think of it like decluttering – we're trying to get everything outside the radical that can be out, leaving only the simplest form inside. To really nail this, you've gotta be comfortable with your basic square roots (like ${\sqrt{4} = 2}$, ${\sqrt{9} = 3}$, ${\sqrt{16} = 4}$, and so on) and how exponents work. Remember, ${\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}}$, which is the golden rule for splitting up radicals. We will need to apply this rule multiple times in this simplification. For example, the number 63 isn't a perfect square, but it does have a perfect square factor, which is crucial for simplifying. Similarly, ${x^3}$ isn't a perfect square either, but it can be expressed as the product of a perfect square and another factor. Keep this concept of extracting perfect square factors in mind as we proceed; it's the key to simplifying radical expressions. So, let’s look closely at how we can break down both the numerical and variable parts of our expression to make the simplification process clearer. We'll be focusing on finding those perfect square factors and using them to our advantage. Think of it as solving a puzzle where each step brings us closer to the neatest, simplest form of our initial expression. This careful approach will not only help us solve this particular problem, but also provide a solid strategy for tackling other similar simplification challenges. Remember, the goal is to express the radical in its most concise form, where no more perfect squares can be extracted. With each step, we're making the expression easier to understand and work with. So, let's start breaking down the components and see what perfect squares we can find hidden within.

Breaking Down the Radicand

So, let's break down the radicand, which is 63x³. Our mission here is to identify any perfect square factors lurking inside. We will address the numerical part (63) and the variable part (x³) separately to make things easier. First, consider the number 63. What are its factors? We can break it down as 63 = 9 * 7. Aha! 9 is a perfect square (since 9 = 3²). This is exactly what we're looking for. So, we can rewrite 63 as 3² * 7. Now, let's tackle the variable part, x³. Remember that exponents tell us how many times a base is multiplied by itself. So, x³ means x * x * x. To find a perfect square factor, we need an even exponent. We can rewrite x³ as x² * x. Here, x² is a perfect square (since it's x times x). By breaking down 63 into 3² * 7 and x³ into x² * x, we've successfully identified the perfect square factors within the radicand. This is a crucial step because we can now use the property ${\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}}$ to separate these perfect squares and simplify our expression. Why is this so important? Because once we isolate the perfect squares, we can take their square roots and move them outside the radical sign. This makes the entire expression simpler and easier to manage. Think of it like sorting through a cluttered space – we're pulling out the items that fit a certain category (perfect squares) so that we can deal with them separately. This strategy not only simplifies the calculation but also makes the process more organized and understandable. So, by recognizing and extracting these perfect square factors, we're setting the stage for the next step in our simplification journey. Let's move on to the next stage and see how we can use these factors to simplify the entire expression.

Extracting Perfect Squares

Alright, we've identified the perfect square factors. Now comes the fun part: extracting them from the radical. We have $-\frac{2}{3} \sqrt{63 x^3}$, which we've rewritten as $-\frac{2}{3} \sqrt{3^2 \cdot 7 \cdot x^2 \cdot x}$. Remember the golden rule: ${\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}}$. This allows us to split the radical into smaller, manageable parts. So, we can rewrite our expression as: $-\frac{2}{3} \cdot \sqrt{3^2} \cdot \sqrt{7} \cdot \sqrt{x^2} \cdot \sqrt{x}$. Now, let's simplify those perfect squares. ${\sqrt{3^2}}$ is simply 3, and ${\sqrt{x^2}}$ is |x| (the absolute value of x, since we want to ensure the result is non-negative). Note that for the purpose of this simplification, we can assume that x is positive, so ${\sqrt{x^2}}$ = x. So, our expression now looks like this: $-\frac{2}{3} \cdot 3 \cdot \sqrt{7} \cdot x \cdot \sqrt{x}$. See how much cleaner it's getting? We've pulled out the perfect squares, leaving the remaining factors inside the radical. But we're not quite done yet. What's the next step? It's time to combine the terms outside the radical and tidy things up. Remember, our goal is to make the expression as simple and clear as possible. This step of extracting perfect squares is critical because it reduces the complexity of the radical, making it easier to work with. By isolating and simplifying these terms, we're essentially decluttering the expression, bringing us closer to the final simplified form. It's like taking out the trash – we're removing the unnecessary complexity, leaving only the essential elements. This methodical approach ensures that we don't miss any steps and that we're handling each component of the expression correctly. So, let's proceed to combine the outside terms and finalize our simplified expression.

Combining Terms

Okay, we're in the home stretch! We've extracted the perfect squares, and now it's time to combine the terms outside the radical. Our expression currently looks like this: $-\frac{2}{3} \cdot 3 \cdot x \cdot \sqrt{7} \cdot \sqrt{x}$. Let's focus on the coefficients first. We have $-\frac{2}{3}$ multiplied by 3. This simplifies beautifully: the 3 in the numerator and the 3 in the denominator cancel each other out, leaving us with -2. So, our expression now becomes: -2 * x * ${\sqrt{7}}$ * ${\sqrt{x}}$. Next, let's look at the radicals. We have ${\sqrt{7}}$ and ${\sqrt{x}}$. Using the property ${\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}}$ in reverse, we can combine these two radicals into a single radical: ${\sqrt{7x}}$. Now, let’s put it all together. We have -2, x, and ${\sqrt{7x}}$. Multiplying these gives us our final simplified form: -2x${\sqrt{7x}}$. And there you have it! We've successfully simplified the original expression. Notice how we took it step by step, breaking down the problem into smaller, manageable parts. This approach is key to tackling any mathematical problem, no matter how intimidating it might seem at first. By combining the terms, we've streamlined the expression, making it as concise as possible. This final step is the culmination of all our hard work – we've untangled the initial complexity and arrived at a clear, simplified answer. This process not only gives us the solution but also reinforces our understanding of radical simplification. Remember, the goal is not just to get the answer, but also to understand the how and why behind each step. So, by combining terms and simplifying, we've completed our journey and arrived at the simplified form of the expression. This methodical approach can be applied to many other mathematical simplifications, providing a solid foundation for more advanced problems.

Final Answer

So, after all that careful work, we've arrived at our final simplified answer: -2x√7x. How cool is that? We started with a seemingly complex expression, $-\frac{2}{3} \sqrt{63 x^3}$, and systematically broke it down using our knowledge of perfect squares and radical properties. We identified the perfect square factors within the radicand (63x³), extracted them, and then combined the remaining terms. This step-by-step approach not only helped us arrive at the correct answer but also reinforced our understanding of the underlying mathematical principles. This isn't just about getting the right answer; it's about the journey of simplification itself. Each step – breaking down the radicand, extracting perfect squares, combining terms – is a mini-victory, building our confidence and skills. And the final answer, -2x√7x, is the ultimate reward, a testament to our persistence and problem-solving abilities. Remember, simplifying radical expressions is a fundamental skill in algebra and beyond. It's like having a superpower that allows you to see through the complexity and reveal the underlying simplicity. And the more you practice, the stronger that superpower becomes. So, don't be afraid to tackle those radical expressions head-on. Break them down, identify the perfect squares, and simplify with confidence. And remember, every simplified expression is a step forward in your mathematical journey. Congrats on mastering this simplification! Now you're ready to tackle the next mathematical challenge with even more skill and confidence. This final simplified form not only represents the solution but also showcases the elegance and efficiency of mathematical operations. It's a reminder that even complex problems can be solved with a methodical approach and a solid understanding of fundamental principles. So, keep practicing, keep simplifying, and keep exploring the wonderful world of mathematics!