Simplifying Radicals Find The Equivalent Expression For Sqrt[4]{(16x^11y^8)/(81x^7y^6)}

Hey guys! Today, we're diving into a fun math problem that involves simplifying radicals. Specifically, we're going to break down the expression $\sqrt[4]{\frac{16 x^{11} y^8}{81 x^7 y^6}}$. This might look intimidating at first glance, but trust me, with a step-by-step approach, it's totally manageable. We'll explore how to tackle each part of the expression and find the equivalent simplified form. So, let's get started and make sure we understand every twist and turn along the way!

Understanding the Problem

Before we jump into solving this, let's make sure we understand what the question is really asking. We need to find an expression that is equivalent to $\sqrt[4]{\frac{16 x^{11} y^8}{81 x^7 y^6}}$. This means we need to simplify the radical as much as possible. Remember, we're working under the assumptions that $x > 0$ and $y \neq 0$, which are important because they help us avoid issues with negative numbers under the radical or division by zero. Simplifying radicals involves breaking down the numbers and variables inside the radical into their simplest forms. We'll be using the properties of exponents and radicals to achieve this. Our goal is to manipulate the expression inside the fourth root, so we can pull out any terms that have a perfect fourth power. This will leave us with a cleaner and simpler expression that's equivalent to the original. Let's break down each component and see how they interact to get the final simplified radical form. Stay with me, and you'll see how each step contributes to the ultimate solution.

Step-by-Step Solution

Okay, let's break this down step-by-step. Simplifying radicals like $\sqrt[4]{\frac{16 x^{11} y^8}{81 x^7 y^6}}$ can seem daunting, but with the right approach, it's totally doable. First, we'll focus on simplifying the fraction inside the radical. This means dealing with the numbers and variables separately. We have $\frac{16}{81}$, $x^{11}$ over $x^7$, and $y^8$ over $y^6$. Let’s tackle these one by one. For the numbers, we’ll look for perfect fourth powers. For the variables, we'll use the exponent rules to simplify. Remember, when dividing terms with the same base, you subtract the exponents. This will help us reduce the expression inside the radical. Then, we’ll apply the fourth root to each term individually. This is where understanding perfect fourth powers becomes crucial. By simplifying each part step by step, we keep the process manageable and reduce the chances of making mistakes. So, let’s dive in and simplify each component methodically!

1. Simplify the Fraction Inside the Radical

The first thing we're going to do is simplify the fraction inside the radical: $\frac{16 x^{11} y^8}{81 x^7 y^6}$. To do this, we'll look at the numerical part and the variable parts separately.

For the numbers, we have $\frac{16}{81}$. We can rewrite 16 as $2^4$ and 81 as $3^4$. So, $\frac{16}{81}$ becomes $\frac{2^4}{3^4}$. This is a crucial step because it sets us up to take the fourth root later on. Recognizing these perfect fourth powers is key to simplifying radicals effectively. For the variables, we'll use the quotient rule for exponents, which says that $\frac{a^m}{a^n} = a^{m-n}$. This rule allows us to simplify expressions with the same base by subtracting the exponents. Now, let’s apply this to the $x$ and $y$ terms in our fraction.

2. Simplify the Variable Terms

Now let's simplify the variable terms. We have $\frac{x^{11}}{x^7}$ and $\frac{y^8}{y^6}$. Using the quotient rule for exponents, we subtract the exponents: $\frac{x^{11}}{x^7} = x^{11-7} = x^4$ and $\frac{y^8}{y^6} = y^{8-6} = y^2$. So now our fraction looks like this: $\frac{2^4 x^4 y^2}{3^4}$. This is much simpler than what we started with! Simplifying the variables using exponent rules is a fundamental part of handling radical expressions. It allows us to reduce complex terms into more manageable ones. This step is particularly important because it prepares the expression for the final simplification under the radical. We've now handled the numerical coefficients and the variable terms, making the next step—applying the fourth root—much easier.

3. Apply the Fourth Root

Now that we've simplified the fraction inside the radical, we can rewrite the original expression as $\sqrt[4]{\frac{2^4 x^4 y^2}{3^4}}$. Remember, the fourth root is the same as raising to the power of $\frac{1}{4}$. So, we can rewrite this as $\left(\frac{2^4 x^4 y^2}{3^4}\right)^{\frac{1}{4}}$. To apply the fourth root, we apply the exponent $\frac{1}{4}$ to each factor in the fraction. This means we have $(2^4)^{\frac{1}{4}}$, $(x^4)^{\frac{1}{4}}$, $(y^2)^{\frac{1}{4}}$, and $(3^4)^{\frac{1}{4}}$. Remember the rule of exponents: $(a^m)^n = a^{mn}$. We'll use this rule to simplify each term.

4. Simplify Each Term

Let's simplify each term by multiplying the exponents. For $(2^4)^{\frac{1}{4}}$, we have $2^{4 \cdot \frac{1}{4}} = 2^1 = 2$. For $(x^4)^{\frac{1}{4}}$, we have $x^{4 \cdot \frac{1}{4}} = x^1 = x$. For $(y^2)^{\frac{1}{4}}$, we have $y^{2 \cdot \frac{1}{4}} = y^{\frac{1}{2}}$. And for $(3^4)^{\frac{1}{4}}$, we have $3^{4 \cdot \frac{1}{4}} = 3^1 = 3$. So, putting it all together, we have $\frac{2xy^{\frac{1}{2}}}{3}$. But we’re not quite done yet! We need to rewrite $y^{\frac{1}{2}}$ in radical form.

5. Rewrite in Radical Form

Finally, let’s rewrite $y^{\frac{1}{2}}$ in radical form. Remember that $y^{\frac{1}{2}}$ is the same as $\sqrt{y}$. So, our simplified expression is $\frac{2x\sqrt{y}}{3}$. This is the fully simplified form of the original expression. We've taken a complex-looking radical and broken it down into something much cleaner and easier to understand. The key here was to tackle each part systematically: simplifying the fraction, applying the exponent rules, and finally, converting back to radical form. This step-by-step approach is super helpful for any radical simplification problem. Now, let's compare our simplified expression with the given options to find the correct answer.

Comparing with the Options

Okay, now that we've simplified the expression to $\frac{2x\sqrt{y}}{3}$, let's compare it with the options provided. We need to find which one matches our simplified form. Looking at the options:

A. $\frac{4 x\left(\sqrt[4]{y^2}\right)}{9}$ B. $\frac{\left.2 \sqrt{4 \sqrt{,^2}}\right)}{3}$ C. $\frac{4 x^2 y}{9}$ D. $\frac{2 x^2 y}{3}$

Our simplified expression, $\frac{2x\sqrt{y}}{3}$, clearly matches option B if we consider that $\sqrt{y}$ is equivalent to $\sqrt[2]{y}$. Option A has a fourth root, and the coefficients and exponents in options C and D don't match our simplified form. Therefore, the correct answer is option B.

Conclusion

So, there you have it! We've successfully simplified the expression $\sqrt[4]{\frac{16 x^{11} y^8}{81 x^7 y^6}}$ to $\frac{2x\sqrt{y}}{3}$. Remember, the key to these problems is breaking them down into smaller, manageable steps. We simplified the fraction, applied exponent rules, took the fourth root, and then rewrote the final answer in radical form. By tackling each part methodically, we avoided getting overwhelmed and found the correct answer. Simplifying radicals might seem tricky at first, but with practice and a clear understanding of the rules, you'll be able to handle these problems with confidence. Keep practicing, and you'll become a pro at simplifying radicals in no time!