Solving Cube Root Expressions: A Step-by-Step Guide
Hey guys! Today, we're diving into a fun math problem involving cube roots and variables. Cube roots might seem a bit intimidating at first, but don't worry! We're going to break it down step by step so it's super easy to understand. We'll tackle this problem: $\sqrt[3]{125 x^{10} y^{13}}+\sqrt[3]{27 x^{10} y^{13}}$, and figure out which of the answer choices (A, B, C, or D) is the correct one. So, grab your pencils, and let's get started!
Understanding the Problem
Before we jump into solving, let’s make sure we understand what the problem is asking. We're given an expression with two terms, both involving cube roots, variables (x and y), and exponents. Our mission, should we choose to accept it (and we do!), is to simplify this expression and match it with one of the provided options. The options are:
- A. $8 x^3 y^4(\sqrt[3]{x y})$
- B. $15 x^8 y^8(\sqrt[3]{x y})$
- C. $15 x^3 y^4(\sqrt[3]{x y})$
- D. $8 x^8 y^8(\sqrt[3]{x y})$
Looks like a fun challenge, right? Remember, the key to solving any math problem is to break it down into smaller, manageable steps. So, let’s do just that!
Breaking Down the Cube Roots
The expression we need to simplify is: $\sqrt[3]{125 x^{10} y^{13}}+\sqrt[3]{27 x^{10} y^{13}}$
Our first step is to simplify each cube root term individually. Let’s start with the first term: $\sqrt[3]{125 x^{10} y^{13}}$
Simplifying the First Term
To simplify a cube root, we need to find factors that are perfect cubes. A perfect cube is a number or expression that can be obtained by multiplying something by itself three times (e.g., 8 is a perfect cube because 2 * 2 * 2 = 8). So, let's look at each part of our term:
- 125: Think, what number times itself three times equals 125? That's right, 5 because 5 * 5 * 5 = 125. So, $\sqrt[3]{125} = 5$.
- x^10: We need to find the largest multiple of 3 that is less than or equal to 10. That's 9 (3 * 3 = 9). So, we can rewrite x^10 as x^9 * x. Then, $\sqrt[3]{x^{10}} = \sqrt[3]{x^9 * x} = x^3 \sqrt[3]{x}$. Remember, we divide the exponent by 3 (9 / 3 = 3) to take it out of the cube root.
- y^13: Similar to x^10, we find the largest multiple of 3 less than or equal to 13, which is 12 (3 * 4 = 12). So, we can rewrite y^13 as y^12 * y. Then, $\sqrt[3]{y^{13}} = \sqrt[3]{y^{12} * y} = y^4 \sqrt[3]{y}$.
Now, let's put it all together:
Awesome! We've simplified the first term. Now, let's move on to the second term.
Simplifying the Second Term
Now let's tackle the second term: $\sqrt[3]{27 x^{10} y^{13}}$
We'll use the same approach as before, breaking it down piece by piece:
- 27: What number times itself three times equals 27? That's 3 (3 * 3 * 3 = 27). So, $\sqrt[3]{27} = 3$.
- x^10: We already know from the previous term that $\sqrt[3]{x^{10}} = x^3 \sqrt[3]{x}$.
- y^13: And we also know that $\sqrt[3]{y^{13}} = y^4 \sqrt[3]{y}$.
Let's combine these:
Fantastic! We've simplified the second term as well.
Combining the Simplified Terms
Now that we've simplified both terms, we can put them back together:
Notice that both terms now have the same radical part: $\sqrt[3]{xy}$. This means we can treat them like like terms and combine them, just like we would combine 5 apples + 3 apples.
To combine like terms, we simply add their coefficients (the numbers in front): 5 + 3 = 8.
So, the simplified expression is:
Finding the Correct Answer
Now, let's compare our simplified expression, $8x3y4\sqrt[3]{xy}$, with the answer choices:
- A. $8 x^3 y^4(\sqrt[3]{x y})$
- B. $15 x^8 y^8(\sqrt[3]{x y})$
- C. $15 x^3 y^4(\sqrt[3]{x y})$
- D. $8 x^8 y^8(\sqrt[3]{x y})$
It's a match! Our simplified expression exactly matches option A.
Conclusion
Therefore, the correct answer is A. $8 x^3 y^4(\sqrt[3]{x y})$
Guys, you did it! We successfully navigated the world of cube roots, variables, and exponents. Remember, the key is to break down complex problems into simpler steps. By simplifying each term individually and then combining like terms, we were able to find the solution with ease.
Key Takeaways:
- Simplifying cube roots involves finding factors that are perfect cubes. Look for numbers or expressions that can be obtained by multiplying something by itself three times.
- When dealing with variables and exponents inside cube roots, divide the exponent by 3 to bring the variable outside the cube root. If the exponent isn't perfectly divisible by 3, you'll have a remainder that stays inside the cube root.
- Like terms can be combined by adding their coefficients. This only works if the terms have the same radical part (in this case, $\sqrt[3]{xy}$).
I hope this step-by-step guide has been helpful. Keep practicing, and you'll become a cube root master in no time! And remember, math can be fun, especially when you break it down and conquer it together.
If you have any other math problems you'd like me to help with, feel free to ask! Until next time, keep learning and keep exploring the amazing world of mathematics! This kind of mathematical problem requires a good understanding of exponents and radicals, so make sure you review those concepts if you're feeling a bit rusty. Understanding how to simplify expressions with radicals is crucial in algebra and beyond. This problem also demonstrates the importance of breaking down complex expressions into smaller, more manageable parts, making the solution process much clearer and less daunting. Keep practicing these skills, and you'll be solving complex algebraic problems like a pro in no time! Good job, everyone! Keep up the great work, and always remember to enjoy the process of learning and problem-solving. Math can be a challenging but incredibly rewarding subject, and every problem you solve helps you build a stronger foundation for future challenges. So, keep exploring, keep questioning, and keep pushing your mathematical boundaries!