Simplifying Cube Roots: A Step-by-Step Solution

Hey guys! Ever stumbled upon a math problem that looks intimidating at first glance? Today, we're diving into simplifying cube roots, specifically the expression $\sqrt[3]{128}-\sqrt[3]{2}$. Don't worry; we'll break it down step by step so you can tackle similar problems with confidence. So, grab your thinking caps, and let's get started!

Understanding Cube Roots

Before we jump into the problem, let's quickly recap what cube roots are. A cube root of a number is a value that, when multiplied by itself three times, gives you the original number. For example, the cube root of 8 is 2 because 2 * 2 * 2 = 8. We denote the cube root using the symbol $\sqrt[3]{}$. Understanding cube roots is the first key step in simplifying expressions like the one we're tackling today. It's essential to recognize perfect cubes – numbers that have whole number cube roots – such as 1, 8, 27, 64, and 125. These numbers often pop up in simplification problems, and knowing them can save you time and effort. In our case, we're dealing with $\sqrt[3]{128}$ and $\sqrt[3]{2}$. At first, it might not be immediately obvious how to simplify these, but that's where our knowledge of perfect cubes comes in handy. We need to find a way to express 128 as a product of a perfect cube and another number. This will allow us to extract the cube root of the perfect cube, making the expression simpler. Think of it like factoring – we're trying to break down the number into its constituent parts, but this time, we're specifically looking for parts that are perfect cubes. This method isn't just applicable to cube roots; it works for other radicals as well, such as square roots and fourth roots. The core idea is always the same: identify the largest perfect power within the radicand (the number inside the root) and extract its root. This process is crucial for simplifying radicals and combining like terms, which is precisely what we'll be doing in our problem. So, with this foundation in place, let's move on to the next step and see how we can apply this to $\sqrt[3]{128}$. We'll be using our understanding of cube roots and perfect cubes to break down the expression and make it easier to work with. Remember, the goal is to simplify, and the key to simplification often lies in recognizing the underlying structure of the numbers we're dealing with.

Simplifying $\sqrt[3]{128}$

Now, let's focus on simplifying $\sqrt[3]{128}$. The trick here is to find the largest perfect cube that divides 128. Think of perfect cubes: 1, 8, 27, 64... Aha! 64 is a perfect cube (4 * 4 * 4 = 64), and it divides 128. In fact, 128 = 64 * 2. So, we can rewrite $\sqrt[3]{128}$ as $\sqrt[3]{64 * 2}$. Remember the rule: $\sqrt[3]{a * b} = \sqrt[3]{a} * \sqrt[3]{b}$. Applying this, we get $\sqrt[3]{64} * \sqrt[3]{2}$. And since $\sqrt[3]{64}$ is 4, we've simplified $\sqrt[3]{128}$ to 4$\sqrt[3]{2}$. Simplifying radicals often involves breaking down the radicand into factors, one of which is a perfect cube (or a perfect square for square roots, etc.). This allows us to extract the root of the perfect cube, leaving a simpler expression. The process of simplifying $\sqrt[3]{128}$ is a prime example of this technique. We identified the largest perfect cube factor of 128, which is 64, and then used the property of radicals to separate the cube root of the product into the product of cube roots. This is a powerful tool in simplifying radical expressions and is a fundamental concept in algebra. Think of it like peeling an onion – we're removing layers until we get to the core. In this case, the layers are the factors of 128, and the core is the simplest form of the radical. Now that we've successfully simplified $\sqrt[3]{128}$, we can move on to the next step, which is substituting this simplified form back into the original expression. This will bring us closer to our final answer and demonstrate the practical application of radical simplification. Remember, the goal is to make the expression as simple as possible, and by breaking down the radicals, we're doing just that. So, let's see how this simplified form helps us in solving the original problem!

Substituting and Solving

Now we know that $\sqrt[3]{128}$ is the same as 4$\sqrt[3]{2}$. Let's plug this back into our original expression: $\sqrt[3]{128}-\sqrt[3]{2}$ becomes 4$\sqrt[3]{2} - \sqrt[3]{2}$. Think of $\sqrt[3]{2}$ as a variable, like 'x'. We're essentially doing 4x - x, which is 3x. So, 4$\sqrt[3]{2} - \sqrt[3]{2}$ simplifies to 3$\sqrt[3]{2}$. And there you have it! We've simplified the expression. Substituting simplified radicals back into the original expression is a crucial step in solving these types of problems. It allows us to combine like terms and further simplify the expression. In our case, after substituting 4$\sqrt[3]{2}$ for $\sqrt[3]{128}$, we were left with an expression that was much easier to handle: 4$\sqrt[3]{2} - \sqrt[3]{2}$. This is where the analogy of treating $\sqrt[3]{2}$ as a variable comes in handy. By thinking of it as a single entity, we can apply the familiar rules of algebra to combine the terms. This is a common technique in simplifying expressions involving radicals, and it's a good way to visualize the process. The key takeaway here is that simplification often involves multiple steps, and substitution is one of the most important. It's like putting the pieces of a puzzle together – each simplified part contributes to the final solution. Now that we've successfully substituted and simplified the expression, we've arrived at our answer: 3$\sqrt[3]{2}$. This is the simplest form of the original expression, and it matches one of the answer choices provided. So, let's recap what we've done and see how this approach can be applied to other problems.

Final Answer

The simplified form of $\sqrt[3]{128}-\sqrt[3]{2}$ is 3$\sqrt[3]{2}$. So the correct answer is (D). Arriving at the final answer involves a combination of understanding cube roots, simplifying radicals, and substituting simplified expressions. In this case, we started by recognizing that 128 could be factored into 64 * 2, where 64 is a perfect cube. This allowed us to simplify $\sqrt[3]{128}$ to 4$\sqrt[3]{2}$. Then, by substituting this back into the original expression, we were able to combine like terms and arrive at the final answer of 3$\sqrt[3]{2}$. This process highlights the importance of breaking down complex problems into smaller, more manageable steps. Each step, from identifying perfect cube factors to substituting simplified expressions, contributes to the overall solution. This approach isn't just limited to simplifying cube roots; it can be applied to a wide range of mathematical problems. The key is to identify the underlying structure of the problem and apply the appropriate techniques to simplify it. So, next time you encounter a seemingly daunting expression, remember the steps we've covered here. Look for perfect powers, simplify radicals, substitute, and combine like terms. With practice, you'll become more confident in your ability to tackle these types of problems. And remember, math isn't just about finding the right answer; it's about understanding the process and developing problem-solving skills that can be applied in many different contexts. So, keep practicing, keep exploring, and keep simplifying!