Simplifying Cube Roots: A Step-by-Step Guide

Hey Plastik Magazine readers! Let's dive into some math, specifically, simplifying cube roots. This problem might look a bit intimidating at first glance, but trust me, we'll break it down step by step and make it super easy to understand. We're going to tackle the expression $\sqrt[3]{125 x^{10} y^{13}}+\sqrt[3]{27 x^{10} y^{13}}$ and figure out which of the multiple-choice answers is the correct simplified form. It's all about understanding the properties of exponents and radicals. We'll be using some basic rules to simplify each term and then combine them. So, grab your notebooks, and let's get started!

Breaking Down the Cube Roots: The First Term

Alright, guys, let's start with the first term: $\sqrt[3]{125 x^{10} y^{13}}$. Our goal here is to simplify this as much as possible. Remember, the cube root of a number is the value that, when multiplied by itself three times, gives you that original number. For instance, the cube root of 8 is 2, because 2 * 2 * 2 = 8. Now, let's break down each part of $\sqrt[3]{125 x^{10} y^{13}}$ separately.

First, the cube root of 125. We know that 5 * 5 * 5 = 125, so $\sqrt[3]{125} = 5$. Easy peasy! Next, let's look at the $x^{10}$ part. When taking a cube root of a variable raised to a power, we divide the exponent by 3. So, for $x^{10}$, we divide 10 by 3. This gives us 3 with a remainder of 1. This means we can take out $x^3$ and leave one $x$ inside the cube root. Therefore, $\sqrt[3]{x^{10}} = x^3\sqrt[3]{x}$. Finally, the cube root of $y^{13}$. Divide 13 by 3, which is 4 with a remainder of 1. So, we can take out $y^4$ and leave one $y$ inside the cube root. Hence, $\sqrt[3]{y^{13}} = y^4\sqrt[3]{y}$. Putting it all together, the simplified form of $\sqrt[3]{125 x^{10} y^{13}}$ is $5x^3y^4\sqrt[3]{xy}$. We have successfully simplified the first part of the expression! The key here is understanding how to apply the cube root to each component: the numerical coefficient, and the variables with their exponents. Remember to divide the exponents of the variables by 3 to find out how much of each variable can be pulled out of the cube root. The remainder becomes the exponent of the variable that stays inside the cube root. This step-by-step approach is crucial.

Detailed Breakdown of the First Term

Let's go into more detail about how we simplified $\sqrt[3]{125 x^{10} y^{13}}$:

1. Cube Root of the Coefficient: We found that $\sqrt[3]{125} = 5$ because 5 * 5 * 5 equals 125.
2. Cube Root of x^10: We divided the exponent 10 by 3. This gives us 3 with a remainder of 1. So, we bring $x^3$ outside the cube root, and $x^1$ (or just x) remains inside. $\sqrt[3]{x^{10}} = x^3\sqrt[3]{x}$.
3. Cube Root of y^13: Dividing the exponent 13 by 3 gives us 4 with a remainder of 1. Hence, $y^4$ comes out of the cube root, and $y^1$ (or just y) stays inside. $\sqrt[3]{y^{13}} = y^4\sqrt[3]{y}$.
4. Combining the Simplified Terms: The simplified form of the first term is $5x^3y^4\sqrt[3]{xy}$. We combined the simplified components to get our final result for the first term of the original expression. Remember, when you're dealing with cube roots, always focus on breaking down each part separately and then putting the pieces back together.

Simplifying the Second Term

Now, let's move on to the second term: $\sqrt[3]{27 x^{10} y^{13}}$. The process here is very similar to what we did for the first term. We're going to break it down piece by piece. First, let's find the cube root of 27. We know that 3 * 3 * 3 = 27, so $\sqrt[3]{27} = 3$. Great!

Next, the $x^{10}$ part is exactly the same as in the first term. When we divided 10 by 3, we got 3 with a remainder of 1. So, we can take out $x^3$ and leave one $x$ inside the cube root. Thus, $\sqrt[3]{x^{10}} = x^3\sqrt[3]{x}$.

Finally, we have $y^{13}$, which, again, is the same as in the first term. Dividing 13 by 3 gives us 4 with a remainder of 1. Thus, we can take out $y^4$ and leave one $y$ inside the cube root. Hence, $\sqrt[3]{y^{13}} = y^4\sqrt[3]{y}$. Putting it all together, the simplified form of $\sqrt[3]{27 x^{10} y^{13}}$ is $3x^3y^4\sqrt[3]{xy}$. Notice the consistency in the process: each term, be it a number or a variable, is dealt with using the same cube root principles. The objective is to make the expression easier to work with.

Detailed Breakdown of the Second Term

Let's break down the simplification of $\sqrt[3]{27 x^{10} y^{13}}$ step-by-step:

1. Cube Root of the Coefficient: We determine that $\sqrt[3]{27} = 3$ since 3 * 3 * 3 equals 27.
2. Cube Root of x^10: As before, we divide the exponent 10 by 3, yielding 3 with a remainder of 1. Thus, we bring $x^3$ outside the cube root, leaving $x^1$ (or simply x) inside. $\sqrt[3]{x^{10}} = x^3\sqrt[3]{x}$.
3. Cube Root of y^13: Dividing the exponent 13 by 3 yields 4 with a remainder of 1. Therefore, $y^4$ comes out of the cube root, and $y^1$ (or y) remains inside. $\sqrt[3]{y^{13}} = y^4\sqrt[3]{y}$.
4. Combining the Simplified Terms: The simplified form of this second term is $3x^3y^4\sqrt[3]{xy}$. Once more, we've combined the simplified components, following the same procedure used in the previous term. This consistent methodology helps keep the process organized and straightforward.

Combining the Simplified Terms

Okay, we've simplified both terms individually. Now, the final step is to combine them. We have $5x^3y^4\sqrt[3]{xy}$ from the first term and $3x^3y^4\sqrt[3]{xy}$ from the second term. Notice something cool, guys? Both terms have the same radical part: $\sqrt[3]{xy}$. This means we can just add the coefficients (the numbers in front of the radical) together. So, we have 5 + 3 = 8. Therefore, the combined expression is $8x^3y^4\sqrt[3]{xy}$. The key here is recognizing that you can only add terms if they have the same radical. It's like adding apples and apples; you don't add apples and oranges. In this case, both terms contained the same cube root expression.

So, the answer is A. $8 x^3 y^4(\sqrt[3]{x y})$. This is because you combine the coefficients of the simplified terms while keeping the radical part the same. It's really that straightforward!

Step-by-Step Combination

Let's clearly outline how we combined the two simplified terms:

1. Identify the Simplified Terms: We previously determined that the first term simplifies to $5x^3y^4\sqrt[3]{xy}$ and the second term simplifies to $3x^3y^4\sqrt[3]{xy}$.
2. Recognize the Common Radical: Both terms include the same radical expression, which is $\sqrt[3]{xy}$. This allows us to combine the terms.
3. Combine the Coefficients: We add the coefficients: 5 + 3 = 8.
4. Write the Combined Expression: The final simplified and combined expression is $8x^3y^4\sqrt[3]{xy}$. This result directly reflects the sum of the original cube root expressions, now in their most simplified form.

Conclusion: The Answer Revealed!

Awesome work, everyone! We've successfully simplified the given expression. By breaking down each term, dealing with the numbers and variables, and then combining the like terms, we found our answer. Remember, the trick to mastering these kinds of problems is to practice regularly. Look at different examples, try solving them yourself, and don't hesitate to go back and review the rules of exponents and radicals. Keep practicing, and you'll become a cube root master in no time! So, the correct answer from the multiple-choice options is:

A. $8 x^3 y^4(\sqrt[3]{x y})$

This is the simplified form of the original expression. Congratulations, you did it!

Hope you enjoyed this lesson. Keep an eye out for more math breakdowns here at Plastik Magazine. Until next time, keep those brains buzzing! Any questions, feel free to drop them in the comments below! We are always here to help you learn and understand new concepts. Keep learning and have fun! The world of mathematics is full of surprises, and with each new problem, you get a chance to sharpen your skills. Cheers!