Simplifying The Expression: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a cool algebra problem that involves simplifying an expression with cube roots. Our goal is to figure out the sum of the given expression and match it with one of the provided options. Let's break it down, step by step, so even if you're not a math whiz, you'll be able to follow along. We will be discussing the given expression and how to simplify it to find the correct answer. The expression we're tackling is: $5x\left(\sqrt[3]{x^2 y}\right) + 2\left(\sqrt[3]{x^5 y}\right)$. Our goal is to find the equivalent expression among the options A, B, C, and D. It might look intimidating at first, but trust me, it's all about understanding the rules of exponents and roots. So, grab your pencils, and let's get started!

Decoding the Expression: Unpacking the Problem

Alright, first things first, let's take a closer look at what we're working with. The expression has two terms: $5x\left(\sqrt[3]{x^2 y}\right)$ and $2\left(\sqrt[3]{x^5 y}\right)$. Both terms involve cube roots, which means we're dealing with the inverse operation of cubing a number (raising it to the power of 3). Remember that the cube root of a number is a value that, when cubed, gives you the original number. For example, the cube root of 8 is 2 because $2^3 = 8$. The variables x and y are inside the cube root, which complicates things a bit, but we'll manage it. Notice that the variables inside the cube roots have different exponents. This is the key to solving the problem! To simplify the expression, we need to try and combine these terms, but before we can do that, we need to make sure the cube roots are simplified as much as possible, or at least have something in common. Let's focus on simplifying each term individually. Remember, the ultimate goal is to see if we can transform the expression to match one of the options provided: A. $7x\left(\sqrt[6]{x^2 y}\right)$, B. $7x^2\left(\sqrt[6]{xy^2}\right)$, C. $7x^2\left(\sqrt[3]{xy^2}\right)$, or D. $7x\left(\sqrt[3]{x^2 y}\right)$. Let's start with the first term.

Simplifying the First Term: $5x\left(\sqrt[3]{x^2 y}\right)$

Let's break down the first term, which is $5x\left(\sqrt[3]{x^2 y}\right)$. There's not much we can do to simplify it further. The cube root of $x^2y$ cannot be simplified without additional information. The $5x$ outside the cube root is already in its simplest form. So, the first term remains as it is: $5x\left(\sqrt[3]{x^2 y}\right)$. It's good to recognize when a term is already in its simplest form. It saves a lot of time and effort. We've got our first term simplified. Keep in mind that sometimes we can't simplify a term as much as we'd like. Now, let's move on to the second term and see if we can do anything there.

Simplifying the Second Term: $2\left(\sqrt[3]{x^5 y}\right)$

Now, let's focus on the second term: $2\left(\sqrt[3]{x^5 y}\right)$. Here, we have $x$ raised to the power of 5 inside the cube root. The key is to recognize that we can rewrite $x^5$ as $x^3 * x^2$ because when you multiply exponents with the same base, you add the powers. So, we can rewrite the second term as: $2\left(\sqrt[3]{x^3 * x^2 * y}\right)$. Now, because the cube root of $x^3$ is $x$, we can take out $x$ from under the cube root, but remember we still have $x^2$ and $y$ inside the root. This simplifies to $2x\left(\sqrt[3]{x^2 y}\right)$. The second term simplifies to $2x\left(\sqrt[3]{x^2 y}\right)$. Excellent! We've managed to simplify the second term. It is very important to remember that when taking out a term from the cube root, it will divide the exponent by 3, so we can only get a full number. Now that we've simplified both terms, we can try to combine them to find the total sum.

Combining the Simplified Terms: Finding the Sum

Now that we've simplified both terms, let's put them back together and see what we get. We had $5x\left(\sqrt[3]{x^2 y}\right)$ from the first term, and $2x\left(\sqrt[3]{x^2 y}\right)$ from the second term. Notice something cool? Both terms now have the same cube root: $\sqrt[3]{x^2 y}$. This means we can combine the terms by adding their coefficients (the numbers in front of the cube root). So, we add 5x and 2x, which gives us $7x\left(\sqrt[3]{x^2 y}\right)$.

Therefore, the sum of the expression $5x\left(\sqrt[3]{x^2 y}\right) + 2\left(\sqrt[3]{x^5 y}\right)$ is $7x\left(\sqrt[3]{x^2 y}\right)$. We have successfully simplified the original expression. Now, we just have to match our answer with one of the provided options.

Matching the Solution: Finding the Correct Answer

Now, let's go back to our options: A. $7x\left(\sqrt[6]{x^2 y}\right)$, B. $7x^2\left(\sqrt[6]{xy^2}\right)$, C. $7x^2\left(\sqrt[3]{xy^2}\right)$, and D. $7x\left(\sqrt[3]{x^2 y}\right)$. Comparing our simplified answer, $7x\left(\sqrt[3]{x^2 y}\right)$, with the options, we can see that it matches perfectly with option D. So, the correct answer is D!

That's all, folks! We've successfully simplified the expression, combined the terms, and matched our result to one of the given options. Math problems like these might seem complex at first, but remember to break them down step by step and apply the rules of exponents and roots. Keep practicing, and you'll get better at it with time. Keep in mind that some questions require a lot more work, but by knowing the basics, you are going to be able to resolve any type of mathematical expression. We hope this explanation has been helpful. Keep up the great work, and see you in the next problem!