Solving For C And D In A Radical Equation

Hey guys! Let's break down how to find the values of $c$ and $d$ that make the equation $\sqrt[3]{162 x^6 y^5}=3 x^2 y(\sqrt[3]{c y^d})$ true. This type of problem might seem intimidating at first, but don't worry, we'll tackle it step by step. We'll focus on simplifying radicals and matching terms to solve for our unknowns. So, grab your thinking caps, and let's dive into the world of radical equations!

Understanding the Problem

Before we jump into solving for the values of $c$ and $d$, it’s super important to get a solid grasp of what the problem is asking. We're given an equation with a cube root on both sides. The left side has a radical expression, $\sqrt[3]{162 x^6 y^5}$, and the right side has a product of terms, including another cube root, $3 x^2 y(\sqrt[3]{c y^d})$. Our mission, should we choose to accept it (and we do!), is to figure out what values of c and d will make this equation a true statement. Think of it like a puzzle where we need to match the pieces perfectly. We need to manipulate the left side of the equation until it looks similar to the right side, and then we can pinpoint the values of c and d. This involves simplifying the radical, identifying perfect cubes, and a bit of algebraic maneuvering. So, let's roll up our sleeves and get started!

Initial Equation Breakdown

Okay, let's dive into the heart of the equation: $\sqrt[3]{162 x^6 y^5}=3 x^2 y(\sqrt[3]{c y^d})$. Our primary goal here is to manipulate the left-hand side to resemble the right-hand side. To achieve this, we'll break down the cube root on the left, simplifying it as much as possible. This involves identifying perfect cubes within the radical. Remember, the cube root of a number is a value that, when multiplied by itself three times, equals that number. For instance, the cube root of 8 is 2 because 2 * 2 * 2 = 8. We'll apply this same logic to the variables and the numerical coefficient within our radical. By extracting these perfect cubes, we'll make the left side of the equation much cleaner and easier to compare with the right side. This is a crucial step in solving for c and d, so let’s get to it and see what we can simplify!

Simplifying the Left Side

Let's tackle the left side of the equation: $\sqrt[3]{162 x^6 y^5}$. Our mission here is to simplify this radical expression. To do this, we'll break down the number 162 and the variables $x^6$ and $y^5$ into their prime factors and look for perfect cubes. Remember, a perfect cube is a number or variable that can be expressed as something raised to the power of 3. For example, $2^3 = 8$, so 8 is a perfect cube. Similarly, $x^3$ is a perfect cube. Our goal is to extract these perfect cubes from under the radical, making the expression simpler and easier to manage. This will involve a bit of number crunching and exponent manipulation, but don't worry, we'll take it one step at a time. Simplifying the left side is key to matching it with the right side of the equation and ultimately solving for c and d. So, let's get our simplification hats on and see what we can find!

Prime Factorization of 162

First, we need to find the prime factorization of 162. This means breaking it down into its prime factors, which are prime numbers that multiply together to give us 162. Let's start by dividing 162 by the smallest prime number, which is 2. 162 divided by 2 is 81. Now, 81 isn't divisible by 2, so we move to the next prime number, 3. 81 divided by 3 is 27. We can divide 27 by 3 again, getting 9. And 9 divided by 3 is 3. So, we've broken 162 down to 2 × 3 × 3 × 3 × 3, which can be written as $2 × 3^4$. This prime factorization is crucial because it helps us identify any perfect cubes within 162. Remember, we're looking for factors that appear three times, as those can be taken out of the cube root. So, let's see how this prime factorization helps us simplify the radical!

Simplifying Variable Terms

Now, let's focus on simplifying the variable terms under the cube root: $x^6$ and $y^5$. When dealing with radicals and exponents, remember the rule that $\sqrt[n]{x^m} = x^{m/n}$. In our case, we're working with a cube root (n = 3), so we're looking for exponents that are divisible by 3. For $x^6$, the exponent 6 is divisible by 3 (6 / 3 = 2), which means $x^6$ is a perfect cube. We can rewrite $\sqrt[3]{x^6}$ as $x^2$. Now, let's tackle $y^5$. The exponent 5 isn't divisible by 3, but we can break $y^5$ down into $y^3 * y^2$. Here, $y^3$ is a perfect cube, and we'll leave $y^2$ under the radical. Simplifying these variable terms is a key step in making the left side of our equation resemble the right side. By extracting the perfect cubes, we're one step closer to solving for c and d. So, with the variables simplified, let's put it all together and see what our simplified left side looks like!

Combining and Simplifying the Radical

Alright, we've done the groundwork – we've found the prime factorization of 162 and simplified the variable terms. Now, let's combine everything and simplify the radical expression $\sqrt[3]{162 x^6 y^5}$. We found that 162 can be written as $2 × 3^4$, $x^6$ is a perfect cube, and $y^5$ can be broken down into $y^3 * y^2$. So, we can rewrite the cube root as follows:

$\sqrt[3]{162 x^6 y^5} = \sqrt[3]{2 × 3^4 × x^6 × y^5}$

Now, let's extract the perfect cubes. We have $3^3$ within $3^4$, $x^6$, and $y^3$ within $y^5$. This gives us:

$\sqrt[3]{2 × 3^3 × 3 × x^6 × y^3 × y^2} = 3x^2y\sqrt[3]{2 × 3 × y^2}$

Simplifying further, we get:

$3x^2y\sqrt[3]{6y^2}$

Wow, we've successfully simplified the left side of the equation! Now it looks much cleaner and more manageable. This simplified form is crucial for comparing it with the right side of the equation and figuring out the values of c and d. So, let's take a moment to appreciate our hard work and then move on to the next step: matching the simplified left side with the right side!

Matching with the Right Side

Okay, guys, this is where the magic happens! We've simplified the left side of the equation to $3x^2y\sqrt[3]{6y^2}$. Now, let's bring back the right side of the equation, which is $3 x^2 y(\sqrt[3]{c y^d})$. Our goal here is to compare these two expressions and figure out the values of c and d that make them equal. Notice that we have the term $3x^2y$ on both sides – that's a good sign! It means we're on the right track. Now, we need to focus on the cube root parts: $\sqrt[3]{6y^2}$ on the left and $\sqrt[3]{c y^d}$ on the right. To make these equal, the values inside the cube roots must be the same. This gives us a direct comparison for c and d. We'll essentially be matching coefficients and exponents to find our unknowns. So, let's put on our detective hats and see what values of c and d we can uncover!

Determining c and d

Alright, let's get down to brass tacks and determine the values of c and d. We have the simplified left side, $3x^2y\sqrt[3]{6y^2}$, and the right side, $3 x^2 y(\sqrt[3]{c y^d})$. As we discussed, we need to match the expressions inside the cube roots. This means we need to find c and d such that:

$\sqrt[3]{6y^2} = \sqrt[3]{c y^d}$

For these two cube roots to be equal, the expressions inside them must be equal. Therefore, we have:

$6y^2 = c y^d$

Now, this is where it becomes pretty straightforward. We can see that the coefficient on the left side is 6, and the exponent of y is 2. By comparing these with the right side, we can directly deduce the values of c and d. c is the coefficient, and d is the exponent of y. So, what do you think? Can you spot the values of c and d? Let's nail this final step and solve for our unknowns!

Solution

Okay, guys, let's wrap this up! By comparing the expressions $6y^2$ and $c y^d$, we can directly identify the values of c and d. The coefficient on the left side is 6, which corresponds to c on the right side. So, c = 6. The exponent of y on the left side is 2, which corresponds to d on the right side. So, d = 2. Therefore, the values that make the equation true are c = 6 and d = 2. This matches option C. We did it! We successfully solved for c and d by simplifying the radical expression, matching terms, and using our knowledge of exponents and coefficients. Give yourselves a pat on the back – you've conquered this radical equation!

Final Answer

So, after all that simplifying, matching, and deducing, we've arrived at our final answer. The values of c and d that make the equation $\sqrt[3]{162 x^6 y^5}=3 x^2 y(\sqrt[3]{c y^d})$ true are:

C. c = 6, d = 2

Great job, everyone! You've navigated through this radical equation like pros. Remember, the key to these types of problems is breaking them down into smaller, manageable steps. Simplifying radicals, identifying perfect cubes, and carefully comparing terms are your best friends in this mathematical adventure. Keep practicing, and you'll become masters of radical equations in no time! And hey, if you ever get stuck, just remember this step-by-step approach, and you'll be solving for c and d like a boss!