Simplifying Expressions: A Math Guide

Hey math enthusiasts! Let's dive into a cool problem involving exponents and simplification. We're going to break down an expression and figure out which ones are equivalent. Get ready to flex those math muscles!

Understanding the Core Problem

Alright, guys, here's what we're working with: We need to find expressions equal to the product of $10x^9$ and $(60x^{-6})^{-1}$. Remember, when we say "product," we mean multiplication. The expression $(60x^{-6})^{-1}$ might look a bit intimidating at first, but we can break it down step-by-step. The key here is to use the rules of exponents. We'll need to remember a few key rules to simplify this expression. For example, a negative exponent means we take the reciprocal. Also, when we multiply terms with the same base, we add the exponents. Let’s get started. First, let's simplify $(60x^{-6})^{-1}$. The negative one exponent means we take the reciprocal: (60x^{-6})^{-1} = rac{1}{60x^{-6}}. Then, we can simplify this further by bringing the $x^{-6}$ up to the numerator, which changes the exponent's sign: rac{1}{60x^{-6}} = rac{x^6}{60}. So, now we can rewrite the original problem as 10x^9 * rac{x^6}{60}. The aim of this problem is to transform the original problem into an easier form, and then compare it with the answers to find the ones that match. This type of simplification is fundamental in algebra and is used extensively in calculus and other areas of mathematics. Think about how these principles might come into play in real-world scenarios – from calculating the growth of a population to understanding compound interest. This knowledge not only helps in solving mathematical problems but also equips you with essential skills for logical reasoning and problem-solving, which are valuable in all aspects of life. Make sure you practice these problems to build your confidence and fluency in solving them. Each problem you solve is a step forward in mastering algebra.

Simplifying the Expression Step-by-Step

Now, let's take the first step. We have $10x^9$ multiplied by rac{x^6}{60}. Let's multiply these two expressions together. We multiply the coefficients (the numbers) and then the variables separately. So, we multiply 10 by rac{1}{60}, which gives us rac{10}{60}. We can simplify this fraction to rac{1}{6}. Then, we multiply $x^9$ by $x^6$. When we multiply terms with the same base (in this case, x), we add the exponents. So, $x^9 * x^6 = x^{9+6} = x^{15}$. Combining these results, our simplified expression becomes rac{1}{6}x^{15} or rac{x^{15}}{6}. This simplification process is critical. Breaking down the problem into smaller, manageable steps reduces the chances of errors and helps us understand the underlying principles better. Mastering this process is important for advanced mathematical concepts. You'll use these principles again and again as you progress in mathematics. Remember, practice is super important. The more you work through these types of problems, the more comfortable and confident you will become. You'll start to recognize patterns and develop a knack for simplifying expressions quickly and accurately. Always double-check your work. It's easy to make a small mistake, but a quick review can save you from a lot of headaches.

Checking the Answer Choices

Now let's go through the answer choices to see which ones match our simplified expression, which is rac{x^{15}}{6}.

Analyzing Each Option

Let’s look at the given options one by one and figure out whether they're equal to our simplified expression. Each option presents a different expression, so we will need to simplify each and see whether it matches. It's like a mini-puzzle where we need to apply the rules of exponents and algebra to verify if each option is correct. The goal is to compare each provided expression against our simplified form, which is rac{x^{15}}{6}. This process not only allows us to determine the correct answers but also reinforces the principles of algebraic manipulation and problem-solving. Make sure to carefully check each step of the process. Remember, accuracy is really important when solving these problems. Always double-check your calculations and simplifications. This will help you find the correct answers. Now, let’s go through each option carefully to determine the correct ones and understand the reasoning behind each choice. This step is crucial for mastering algebra. Here is how we will approach this:

• Option 1: rac{10x^9}{60x^{-3}}: When dividing terms with the same base, you subtract the exponents. So we have: rac{10}{60} * rac{x^9}{x^{-3}} = rac{1}{6} * x^{9 - (-3)} = rac{1}{6} * x^{12}. This is not equal to rac{x^{15}}{6}.
• Option 2: rac{10x^9}{60x^6}: Simplify the coefficients: rac{10}{60} = rac{1}{6}. Then, subtract the exponents: rac{x^9}{x^6} = x^{9-6} = x^3. Therefore, this expression simplifies to rac{1}{6} * x^3, which is not equal to rac{x^{15}}{6}.
• Option 3: rac{x^{15}}{6}: This expression exactly matches our simplified expression. This is the correct answer. This option is already in the simplified form that we derived earlier, so we can see that this is a valid solution.

Identifying the Correct Options

Based on our analysis, only one option is correct. After we simplify, we find that rac{x^{15}}{6} is our answer. So, the correct answer is the expression that simplifies to rac{x^{15}}{6}. It’s all about applying the rules systematically and carefully. This process demonstrates how algebraic principles are applied. The ability to manipulate and simplify expressions is a fundamental skill in mathematics. This skill is critical for solving more complex problems. Regular practice helps you to hone these abilities. Keep going, and you'll become more confident in your algebra skills.

Conclusion

Alright, guys, that's a wrap! We've successfully simplified the expression and identified the equivalent one. This exercise really highlights the importance of understanding and correctly applying the rules of exponents. Keep practicing, and you'll get the hang of it in no time. Always remember to break down complex problems into smaller, more manageable steps, and double-check your work. You've got this!