Simplify Radical Expressions: Step-by-Step Guide

Hey there, math enthusiasts! Let's dive into simplifying this radical expression: $\sqrt{\frac{128 x^5 y^6}{2 x^7 y^5}}$. Our goal is to find an equivalent expression from the given options. Don't worry, we'll break it down step by step to make sure everyone's on board. Before we start, remember the golden rules: $x > 0$ and $y > 0$. This means we're dealing with positive numbers, which simplifies things quite a bit.

Step 1: Simplify the Fraction Inside the Square Root

The first thing we need to tackle is the fraction inside the square root. We've got $\frac{128 x^5 y^6}{2 x^7 y^5}$. Let's simplify the numerical part and then handle the variables. So, when you first look at the expression, it might seem a bit intimidating with all those exponents and variables. But trust me, we're going to break it down into manageable chunks, and by the end, you'll be like, "Oh, that's all there is to it?" We'll start by simplifying the numerical part of the fraction, which is $\frac{128}{2}$. What's 128 divided by 2, guys? That's right, it's 64. So, we can rewrite our fraction as $\frac{64 x^5 y^6}{x^7 y^5}$. Now, let's move on to the variables. Remember the rule for dividing exponents with the same base? You subtract the exponents. For $x$, we have $x^5$ in the numerator and $x^7$ in the denominator. So, we subtract the exponents: $5 - 7 = -2$. That means we'll have $x^{-2}$, which is the same as $\frac{1}{x^2}$. For $y$, we have $y^6$ in the numerator and $y^5$ in the denominator. Subtracting the exponents, we get $6 - 5 = 1$. So, we have $y^1$, which is just $y$. Putting it all together, we've simplified the fraction inside the square root to $\frac{64y}{x^2}$. See? We're making progress already!

Numerical Part

Let's start with the numbers: $\frac{128}{2}$. This simplifies to 64. Easy peasy!

Variable Part: x

Now, let's look at the $x$ terms: $\frac{x^5}{x^7}$. Remember the rule for dividing exponents with the same base? You subtract the exponents: $x^{5-7} = x^{-2}$. A negative exponent means we can rewrite this as $\frac{1}{x^2}$. So, when you see those exponents, don't sweat it! Just remember the rules. Dividing exponents with the same base? Subtract 'em. A negative exponent? Flip it and make it positive. These are the little tricks that make simplifying expressions like this so much easier. And once you get the hang of it, you'll be cruising through these problems like a pro. We've tackled the numerical part and the x part, and now we're onto the y's. We're building up our simplified fraction piece by piece, and it's all coming together nicely. So, let's keep the momentum going and see what we can do with those y's!

Variable Part: y

Next up, the $y$ terms: $\frac{y^6}{y^5}$. Again, subtract the exponents: $y^{6-5} = y^1$, which is just $y$. Fantastic!

Putting It Together

Combining these simplifications, our fraction becomes:

$\frac{64y}{x^2}$

Step 2: Apply the Square Root

Now we have $\sqrt{\frac{64 y}{x^2}}$. Remember, the square root applies to both the numerator and the denominator. So, we can think of this as $\frac{\sqrt{64y}}{\sqrt{x^2}}$. Applying the square root might seem like the trickiest part, but it's really just about knowing your squares and using a few key properties of square roots. First, let's think about the numerator: $\sqrt{64y}$. We can actually split this up into $\sqrt{64} \cdot \sqrt{y}$. Why? Because the square root of a product is the product of the square roots. This is a super handy trick to remember. Now, what's the square root of 64? If you're thinking 8, you're spot on! So, $\sqrt{64}$ is 8, and we have $8\sqrt{y}$ in the numerator. Next, let's look at the denominator: $\sqrt{x^2}$. This one's pretty straightforward. The square root of $x^2$ is just $x$, since the square root and the square cancel each other out. So, now we have $\frac{8\sqrt{y}}{x}$. We've simplified the expression inside the square root, and we've applied the square root to both the numerator and the denominator. We're almost there, guys! We just need to take a look at our answer choices and see which one matches up with what we've got. So, let's do that now.

Square Root of the Numerator

Let's break down the numerator: $\sqrt{64y}$. We can split this into $\sqrt{64} \cdot \sqrt{y}$. The square root of 64 is 8, so we have $8\sqrt{y}$. Remember that $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$. This property is super useful!

Square Root of the Denominator

The denominator is $\sqrt{x^2}$. The square root and the square cancel each other out, leaving us with $x$. So, whenever you see a variable squared inside a square root, you can just take the variable itself as the result. It's like a magic trick, but it's math!

Putting It Together Again

Our expression now looks like this:

$\frac{8 \sqrt{y}}{x}$

Step 3: Match the Result with the Options

Our simplified expression is $\frac{8 \sqrt{y}}{x}$. Looking at the options, we can see that this matches option D.

So, let's recap what we've done. We started with a seemingly complex expression, $\sqrt{\frac{128 x^5 y^6}{2 x^7 y^5}}$, and we systematically simplified it step by step. We divided the numerical coefficients, subtracted the exponents for the variables, and then applied the square root to both the numerator and the denominator. And all that hard work paid off, because we arrived at the simplified expression $\frac{8 \sqrt{y}}{x}$, which matches option D. Remember, guys, math isn't about memorizing formulas or blindly following procedures. It's about understanding the underlying principles and applying them in a logical way. So, keep practicing, keep exploring, and keep having fun with it!

Final Answer

The equivalent expression is:

D. $\frac{8 \sqrt{y}}{x}$

So, there you have it! We've successfully navigated through the simplification of this radical expression. Remember, the key is to break it down into smaller, manageable steps. Simplify the fraction first, then apply the square root, and finally, match your result with the given options. You've got this!