Simplifying Fourth Root Expressions A Step-by-Step Guide

Hey guys! Let's dive into a fun math problem together. We've got this expression with a fourth root, and our mission, should we choose to accept it, is to simplify it. It looks a bit intimidating at first, but don't worry, we'll break it down step by step and make it super clear. So, grab your thinking caps, and let's get started!

The Challenge: Simplifying $\sqrt[4]{\frac{24 x^6 y}{128 x^4 y^5}}$

The expression we need to simplify is $\sqrt[4]{\frac{24 x^6 y}{128 x^4 y^5}}$. This looks like a mouthful, right? But trust me, we can tame this beast. Our goal is to find an equivalent expression among the options provided. To do this effectively, we're going to use the magic of simplifying radicals and exponents. Remember, the key to tackling these problems is to break them down into smaller, manageable chunks. We'll start by simplifying the fraction inside the fourth root, then we'll deal with the exponents and the radical itself. Keep your eyes peeled, and let's see how this unfolds!

Step-by-Step Simplification

Okay, let's break down this problem into bite-sized pieces. First, we're going to simplify the fraction inside the fourth root. We have $\frac{24 x^6 y}{128 x^4 y^5}$. The numbers 24 and 128 can be simplified by finding their greatest common divisor, which is 8. Dividing both by 8, we get $\frac{3}{16}$. Now, let's tackle the variables. We have $x^6$ divided by $x^4$, and using the rule of exponents (subtracting exponents when dividing), we get $x^{6-4} = x^2$. For the y terms, we have $y$ divided by $y^5$, which gives us $y^{1-5} = y^{-4}$. Remember, a negative exponent means we can move the term to the denominator, so $y^{-4}$ becomes $\frac{1}{y^4}$.

Putting it all together, the fraction simplifies to $\frac{3x^2}{16y^4}$. Now we have $\sqrt[4]{\frac{3x^2}{16y^4}}$. We're getting closer! Next, we'll deal with the fourth root. The goal here is to see what we can pull out of the radical. Remember, we're looking for factors that appear four times (since it's a fourth root). This step is like an archeological dig – we're excavating the perfect squares hidden within the expression.

Taming the Fourth Root

Now that we've simplified the fraction inside the fourth root, let's tackle the root itself. We have $\sqrt[4]{\frac{3x^2}{16y^4}}$. We can rewrite the fourth root of a fraction as the fraction of the fourth roots: $\frac{\sqrt[4]{3x^2}}{\sqrt[4]{16y^4}}$. This makes it easier to handle each part separately. Let's start with the denominator. We need to find the fourth root of $16y^4$. We know that $16 = 2^4$, so $\sqrt[4]{16} = 2$. And $\sqrt[4]{y^4} = y$ (since the fourth root and the fourth power cancel each other out). So, the denominator simplifies to $2y$.

Now, let's look at the numerator: $\sqrt[4]{3x^2}$. We can't simplify the fourth root of 3 any further because 3 doesn't have any factors that appear four times. For $x^2$, since the exponent 2 is less than the index 4 of the root, we can't pull any whole x terms out of the radical. So, $\sqrt[4]{3x^2}$ stays as it is. Putting the numerator and denominator together, we get $\frac{\sqrt[4]{3x^2}}{2y}$. And just like that, we've simplified the entire expression! We've gone from a complex-looking radical expression to something much cleaner and easier to understand. It's like we've transformed a tangled mess into a clear, elegant solution. Awesome, right?

Matching the Equivalent Expression

Alright, we've simplified our expression to $\frac{\sqrt[4]{3x^2}}{2y}$. Now, let's compare this to the answer choices provided. Remember, our goal is to find the expression that is equivalent to what we've simplified.

• A. $\frac{\sqrt[4]{3}}{2 x^2 y}$: This one has an $x^2$ term in the denominator that we don't have, and it's missing the $x^2$ inside the fourth root in the numerator. So, it's not a match.
• B. $\frac{x(\sqrt[4]{3})}{4 y^2}$: This option has an x term outside the radical, which we don't have in our simplified expression, and it also has a $y^2$ in the denominator instead of a $y$. So, it's not the correct answer either.
• C. $\frac{\sqrt[4]{3}}{4 x y^2}$: This option is missing the $x^2$ term under the fourth root in the numerator, and it has an $xy^2$ term in the denominator, which doesn't match our $2y$. So, it's not the right one.
• D. $\frac{\sqrt[4]{3 x^2}}{2 y}$: Bingo! This expression matches perfectly with our simplified expression. It has the fourth root of $3x^2$ in the numerator and $2y$ in the denominator. This is the one!

So, the expression equivalent to $\sqrt[4]{\frac{24 x^6 y}{128 x^4 y^5}}$ is $\frac{\sqrt[4]{3 x^2}}{2 y}$. We did it! We successfully simplified the expression and found the matching equivalent one. Give yourselves a pat on the back, you've earned it!

Conclusion: The Power of Simplification

Wow, we've been on quite the journey, haven't we? We started with a complex-looking fourth root expression and, through careful simplification, we found its equivalent form. The key takeaways here are the power of breaking down problems into smaller steps, simplifying fractions, applying exponent rules, and understanding how to work with radicals. These are powerful tools in your math arsenal, and the more you practice, the more comfortable you'll become using them. Remember, math isn't about memorizing formulas; it's about understanding the process and applying the concepts. So keep practicing, keep exploring, and most importantly, keep having fun with math! And hey, if you ever stumble upon another tricky problem, you know where to find us. We're always up for a mathematical adventure. Until next time, keep those brains buzzing!