Simplifying Cube Roots: A Step-by-Step Guide
Hey guys! Ever stumbled upon a cube root that looks like it needs a little tidying up? You're not alone! Cube roots, especially those involving fractions, can seem a bit intimidating at first. But don't worry, we're going to break it down and make simplifying them super easy. We'll take a look at the expression $\sqrt[3]{\frac{4 x}{5}}$ as an example. We'll walk through each step, so you'll be simplifying like a pro in no time. Let's dive in!
Understanding Cube Roots
Before we jump into simplifying, let's quickly recap what cube roots are all about. Think of it this way: the cube root of a number is the 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. Makes sense, right? When dealing with fractions and variables inside a cube root, the same principle applies, but we need to use a few extra tricks to get them into their simplest form. We often aim to get rid of any fractions inside the radical and ensure the denominator is a nice, whole number. This is where the fun begins!
Breaking Down the Expression
Okay, let's get our hands dirty with our expression: $\sqrt[3]\frac{4 x}{5}}$. The first thing we want to do is tackle that fraction inside the cube root. It's like having a party and everyone's cramped in one room – we need to spread things out! A key property of radicals (like our cube root) is that we can separate the root of a fraction into the fraction of the roots. In simpler terms, the cube root of a fraction is the same as the cube root of the numerator divided by the cube root of the denominator. So, we can rewrite our expression as}{\sqrt[3]{5}}$. See? We've given the numerator and denominator their own space! But we're not done yet; we've still got some simplifying to do.
Rationalizing the Denominator
Now, let's talk about "rationalizing the denominator." It sounds super technical, but it's really just a fancy way of saying we want to get rid of any radicals (like our cube root) in the bottom of the fraction. Why? Because mathematicians like things neat and tidy, and having a radical in the denominator is considered a bit messy. So, how do we do it? Well, we need to think about what would make that cube root in the denominator disappear. Remember, we need a number that, when multiplied by itself three times, gives us a perfect cube (like 8, 27, 64, etc.). Currently, we have $\sqrt[3]5}$ in the denominator. To make this a perfect cube, we need to multiply it by something that will result in a power of 3 inside the cube root. We need to multiply 5 by 5 * 5 = 25 to get 125, which is 5 cubed! But here's the catch$.
Multiplying by the Magic Factor
Let's do it! We're multiplying both the top and bottom of our fraction by $\sqrt[3]25}$. This gives us}\sqrt[3]{5}} * \frac{\sqrt[3]{25}}{\sqrt[3]{25}}$. Now, we multiply the numerators together and the denominators together. Remember, when multiplying radicals with the same index (in this case, a cube root), we can multiply the numbers inside the radicals * \sqrt[3]25} = \sqrt[3]{4x * 25} = \sqrt[3]{100x}$ Denominator * \sqrt[3]25} = \sqrt[3]{5 * 25} = \sqrt[3]{125}$ So, our expression now looks like this}{\sqrt[3]{125}}$. We're getting closer! Notice anything special about that denominator?
Simplifying the Denominator
Aha! The denominator, $\sqrt[3]125}$, is a perfect cube! We know that 125 is 5 cubed (5 * 5 * 5 = 125), so the cube root of 125 is simply 5. This means we can replace $\sqrt[3]{125}$ with 5 in our expression. Our fraction now looks like this}{5}$. Look at that! We've successfully rationalized the denominator – no more cube root down there! But before we declare victory, let's double-check if we can simplify the numerator any further.
Checking the Numerator
Let's take a closer look at the numerator: $\sqrt[3]{100x}$. We need to see if there are any perfect cube factors hiding inside 100. Remember, we're looking for factors that are perfect cubes, like 8, 27, 64, etc. The prime factorization of 100 is 2 * 2 * 5 * 5, or 2^2 * 5^2. Sadly, there are no perfect cube factors here. We don't have three 2's or three 5's. Also, the variable x is just x to the power of 1, so we can't take a cube root of that either. This means that $\sqrt[3]{100x}$ is already in its simplest form. Woohoo!
The Final Simplified Form
Alright, guys, we've done it! We took that scary-looking cube root expression and simplified it step by step. We separated the fraction, rationalized the denominator, and checked for any further simplifications in the numerator. The final simplified form of $\sqrt[3]\frac{4 x}{5}}$ is}{5}$. Isn't that satisfying? Remember, the key to simplifying cube roots (and other radicals) is to break things down, look for perfect cubes, and don't be afraid to tackle each step one at a time. With a little practice, you'll be a cube rootSimplifier in no time! Keep up the great work!
So, the correct answer is:
D. $\frac{\sqrt[3]{100 x}}{5}$