Simplifying Radicals A Step-by-Step Guide To Cube Roots

Hey everyone! Today, we're diving into the world of simplifying radical expressions, specifically cube roots. We'll be tackling the expression $\sqrt[3]{686x^4y^7}$, breaking it down step by step to arrive at the simplest form. This might seem daunting at first, but trust me, it's totally manageable once you grasp the underlying principles. So, let's roll up our sleeves and get started! We'll cover everything from prime factorization to identifying perfect cubes and extracting them from the radical. By the end of this guide, you'll not only be able to simplify this particular expression but also have a solid foundation for tackling similar problems. Think of it as unlocking a superpower in algebra! Let's embark on this mathematical adventure together, guys!

Understanding Cube Roots

Before we jump into the problem itself, let's quickly recap what cube roots are all about. A cube root of a number is a value that, when multiplied by itself three times, gives you the original number. For instance, the cube root of 8 is 2 because 2 * 2 * 2 = 8. We denote the cube root using the radical symbol $\sqrt[3]{ }$, with the little '3' indicating it's a cube root rather than a square root (which has an implied '2' in the index). When simplifying cube roots, our goal is to identify perfect cubes within the radicand (the expression under the radical) and extract them. A perfect cube is a number or expression that can be obtained by cubing an integer or a variable. This is a crucial concept that forms the bedrock of simplifying radical expressions. To truly master simplifying cube roots, you need to be comfortable identifying perfect cubes. Think of numbers like 8, 27, 64, 125, and so on, as well as expressions like $x^3$, $y^6$, and $z^9$. These are your key ingredients for successfully simplifying radicals. Remember, we are looking for groups of three identical factors, not pairs like we do with square roots.

Understanding the properties of exponents is also key here. Specifically, recall that $(a*b)^n = a^n * b^n$ and $\sqrt[n]{a*b} = \sqrt[n]{a} * \sqrt[n]{b}$. These properties allow us to break down complex radicals into simpler components, making the simplification process much more manageable. For example, we can separate the radical into individual terms involving the numerical coefficient and the variables. This tactic is especially useful when dealing with expressions containing both numbers and variables raised to different powers, as in our case with $\sqrt[3]{686x^4y^7}$. It's like having a set of tools that let you disassemble a complicated machine into its individual parts for easier inspection and repair. Once you understand the fundamental concept of perfect cubes and how they interact with radical expressions, you're well on your way to mastering the art of simplifying cube roots. So, with this in mind, let's move on to the first step in simplifying our expression.

Step 1: Prime Factorization of 686

The first step in simplifying $\sqrt[3]{686x^4y^7}$ is to break down the numerical coefficient, 686, into its prime factors. This means expressing 686 as a product of prime numbers – numbers that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11...). Prime factorization helps us identify any perfect cubes hidden within the number. Think of it like detective work, where we're uncovering the building blocks that make up 686. We start by dividing 686 by the smallest prime number, 2, which gives us 343. Then, we try dividing 343 by the next smallest prime number, 3, but it doesn't divide evenly. We continue checking larger prime numbers, and we find that 343 is divisible by 7, resulting in 49. Finally, 49 is divisible by 7, giving us 7. And, of course, 7 is divisible by 7, resulting in 1. So, the prime factorization of 686 is 2 * 7 * 7 * 7, or more compactly, 2 * $7^3$. Do you see that perfect cube hiding in there? $7^3$ is a clear indicator that we're on the right track! Prime factorization is not just a mathematical technique; it's a way to reveal the underlying structure of a number. By breaking down 686 into its prime factors, we can clearly see the presence of the perfect cube $7^3$. This is a crucial step because it allows us to extract the cube root of $7^3$, which is simply 7. Without prime factorization, we might miss this opportunity and struggle to simplify the expression effectively. In essence, it's like having a magnifying glass that helps us see the smaller details within a larger number, making the task of simplification much easier. Now that we have the prime factorization of 686, let's move on to the next step, which involves dealing with the variables in our expression.

Step 2: Simplifying the Variable Terms ($x^4$ and $y^7$)

Now that we've tackled the numerical coefficient, let's turn our attention to the variable terms, $x^4$ and $y^7$. Our goal here is to express these terms as products of perfect cubes and any remaining factors. Remember, we're looking for exponents that are multiples of 3, as these can be easily extracted from the cube root. For $x^4$, we can rewrite it as $x^3 * x$. Notice that $x^3$ is a perfect cube, as it's the result of cubing $x$ ($x * x * x = x^3$). The remaining factor, $x$, will stay under the radical. It's like sorting your books into shelves – you want to group the ones that fit the shelf perfectly (the perfect cubes) and leave the odd ones out (the remaining factors). Similarly, for $y^7$, we can rewrite it as $y^6 * y$. Here, $y^6$ is a perfect cube because it's equivalent to $(y^2)^3$ ($y^2 * y^2 * y^2 = y^6$). Again, the remaining factor, $y$, will stay under the radical. Simplifying variable terms involves recognizing the largest multiple of 3 that is less than the exponent and then separating the term accordingly. This process allows us to isolate the perfect cube component, making it easier to extract from the radical. Think of it as carefully dissecting the variable terms to reveal their cube root potential. By strategically breaking down $x^4$ and $y^7$, we've identified the perfect cubes $x^3$ and $y^6$, which will play a crucial role in our final simplified expression. This step is all about strategic rearrangement, setting the stage for extracting the cube roots of these perfect cube components. With the variables simplified, we're now ready to piece everything together and see the simplified expression take shape. So, let's move on to the exciting part – combining our findings!

Step 3: Combining and Simplifying the Radical

Alright, we've done the groundwork – prime factorization of 686 and simplification of the variable terms. Now comes the moment of truth: combining everything under the radical and simplifying it. Let's recap what we have so far: $\sqrt[3]{686x^4y^7}$ can be rewritten as $\sqrt[3]{2 * 7^3 * x^3 * x * y^6 * y}$. Remember the property $\sqrt[n]{a*b} = \sqrt[n]{a} * \sqrt[n]{b}$? We're going to use this to separate the perfect cubes from the remaining factors. We can rewrite the expression as $\sqrt[3]{7^3} * \sqrt[3]{x^3} * \sqrt[3]{y^6} * \sqrt[3]{2 * x * y}$. Now, we can take the cube root of the perfect cubes. The cube root of $7^3$ is simply 7. The cube root of $x^3$ is $x$. And the cube root of $y^6$ is $y^2$ (since $(y^2)^3 = y^6$). So, we have $7 * x * y^2 * \sqrt[3]{2xy}$. This is our simplified expression! We've successfully extracted all the perfect cubes from the radical, leaving behind only the irreducible factors. Think of this step as the grand finale of our simplification journey. We've carefully gathered all the pieces – the prime factors, the simplified variable terms – and now we're assembling them to create the final product. The process of combining and simplifying the radical involves strategically applying the properties of radicals and cube roots, allowing us to neatly separate the terms that can be extracted from those that must remain under the radical sign. It's like conducting an orchestra, where each instrument (the individual factors) plays its part in creating a harmonious whole (the simplified expression). We've transformed a seemingly complex expression into a much more manageable form. This is the power of simplification!

Final Answer

Therefore, the simplified form of $\sqrt[3]{686x^4y^7}$ is $7xy^2\sqrt[3]{2xy}$. You nailed it! We've successfully navigated the world of cube roots, broken down a complex expression, and arrived at our simplified answer. Remember, the key is to break down the problem into smaller, manageable steps: prime factorization, simplifying variable terms, and combining the results. Each step builds upon the previous one, leading us to the final solution. This process not only allows us to simplify radical expressions but also deepens our understanding of mathematical principles. By understanding these principles, we are empowered to tackle a wide range of problems. So, keep practicing, keep exploring, and keep simplifying! The world of mathematics is full of exciting challenges, and you now have the skills to conquer them. Keep up the great work, guys!

• $7 x y^2 \sqrt[3]{2 x y}$
• $7 x^2 y \sqrt[3]{2 x y^2}$
• $7 x y \sqrt[3]{x y^2}$
• $7 x y^3 \sqrt[3]{2 x y}$