Solving For C And D In Cube Root Equations

Hey guys! Let's dive into a fun math problem where we need to figure out the values of $c$ and $d$ that make a cube root equation true. This type of problem often pops up in algebra, and it's a great way to sharpen our skills with radicals and exponents. The equation we're tackling today is $\sqrt[3]{162 x^6 y^5}=3 x^2 y\left(\sqrt[3]{c y^d}\right)$. So, let's break it down step by step and find out how to solve it. We'll focus on simplifying the cube root, identifying common factors, and then matching the coefficients and exponents. Stick with me, and you'll see it's not as intimidating as it looks! Understanding these concepts will not only help you solve this specific problem but also give you a solid foundation for tackling similar algebraic challenges. Let's get started and make math a little less mysterious together.

Understanding the Problem

Before we jump into solving for $c$ and $d$, let's make sure we really understand what the problem is asking. We've got this equation: $\sqrt[3]{162 x^6 y^5}=3 x^2 y\left(\sqrt[3]{c y^d}\right)$. Our mission, should we choose to accept it (and we do!), is to find the values of $c$ and $d$ that make this equation true. Think of it like a puzzle where we need to find the right pieces that fit perfectly. On the left side, we have a cube root containing a mix of numbers and variables. On the right side, we have a similar expression, but with unknowns $c$ and $d$. Our job is to manipulate the left side so it looks like the right side, and then we can easily see what $c$ and $d$ must be. This involves simplifying radicals, understanding exponents, and a bit of algebraic maneuvering. It's like being a math detective, uncovering clues until we crack the case! So, let's put on our detective hats and get to work.

Simplifying the Cube Root

Okay, the first step in our math detective work is to simplify the cube root on the left side of the equation: $\sqrt[3]{162 x^6 y^5}$. To do this, we need to break down the number and variables inside the cube root and see if we can pull out any perfect cubes. Let's start with 162. We need to find its prime factorization. 162 can be written as $2 imes 81$, and 81 is $3^4$. So, $162 = 2 imes 3^4$. Now, let's look at the variables. We have $x^6$ and $y^5$. Remember, when taking the cube root, we're looking for exponents that are multiples of 3. For $x^6$, the exponent 6 is a multiple of 3, so we can easily take its cube root. For $y^5$, we can rewrite it as $y^3 imes y^2$, where $y^3$ is a perfect cube. Now, let's rewrite the entire cube root: $\sqrt[3]{162 x^6 y^5} = \sqrt[3]{2 imes 3^4 imes x^6 imes y^3 imes y^2}$. We can separate this into $\sqrt[3]{3^3 x^6 y^3} imes \sqrt[3]{2 imes 3 imes y^2}$. Taking the cube root of the first part gives us $3x^2y$. So, the simplified form is $3 x^2 y \sqrt[3]{2 imes 3 imes y^2}$ which simplifies further to $3x^2y \sqrt[3]{6y^2}$.

Matching the Equation

Now that we've simplified the left side of the equation, let's bring back the original equation and see how our simplified form helps us find $c$ and $d$. Our simplified left side is $3x^2y \sqrt[3]{6y^2}$, and the original equation is $\sqrt[3]{162 x^6 y^5}=3 x^2 y\left(\sqrt[3]{c y^d}\right)$. We've already transformed the left side into $3x^2y \sqrt[3]{6y^2}$. Now, we can directly compare this to the right side of the equation, which is $3 x^2 y\left(\sqrt[3]{c y^d}\right)$. Notice that we have $3x^2y$ on both sides, which is great! This means we can focus on the cube root parts: $\sqrt[3]{6y^2}$ and $\sqrt[3]{cy^d}$. For these two expressions to be equal, the values inside the cube roots must be the same. So, we need to match $6y^2$ with $cy^d$. This is where it becomes pretty straightforward. We can see that $c$ corresponds to 6, and $y^d$ corresponds to $y^2$. Therefore, by matching the coefficients and the exponents, we can determine the values of $c$ and $d$. It's like fitting the last pieces of the puzzle into place, and we're almost there!

Determining the values of c and d

Alright, let's nail down the values of $c$ and $d$. We've simplified the equation and matched the terms, and now we're in the home stretch. We have $\sqrt[3]{6y^2}$ on the left side and $\sqrt[3]{cy^d}$ on the right side. To make these two expressions equal, the terms inside the cube roots must be identical. This means that $6y^2$ must be equal to $cy^d$. Now, let's break this down. The coefficient of the $y^2$ term on the left side is 6, and on the right side, it's $c$. So, for the expressions to be equal, $c$ must be 6. That's one variable down! Next, let's look at the exponents of $y$. On the left side, the exponent of $y$ is 2, and on the right side, it's $d$. Again, for the expressions to be equal, $d$ must be 2. So, we've found that $c = 6$ and $d = 2$. We've solved the puzzle! We've successfully determined the values of $c$ and $d$ that make the equation true. It's like cracking a code, and we did it by carefully simplifying, matching terms, and using our algebraic skills. High five!

Final Answer

So, after all our hard work simplifying, comparing, and matching, we've arrived at the final answer. The values of $c$ and $d$ that make the equation $\sqrt[3]{162 x^6 y^5}=3 x^2 y\left(\sqrt[3]{c y^d}\right)$ true are $c = 6$ and $d = 2$. This corresponds to option C. Isn't it satisfying when you solve a problem step by step and get to the right solution? We started with a seemingly complex equation, broke it down into manageable parts, and used our knowledge of cube roots and exponents to find the answer. Remember, guys, the key to tackling these kinds of problems is to simplify as much as possible, look for common factors, and match the terms carefully. And now, you've got another tool in your math toolkit! You're well-equipped to handle similar challenges in the future. Keep practicing, and math will become less daunting and even (dare I say it?) fun!