Simplifying Exponents: Find Equivalent Expression

Hey there, math enthusiasts! Today, we're diving into the world of exponents to simplify a seemingly complex expression. Our mission is to figure out which option is equivalent to $\frac{267^5}{267^2}$. This kind of problem often pops up in algebra, and mastering it can seriously boost your math skills. We'll break it down step-by-step, so you'll not only get the answer but also understand the why behind it. Let's get started and unravel this exponential puzzle together! Understanding the rules of exponents is crucial for simplifying expressions like this one. Exponents, at their core, are a shorthand way of writing repeated multiplication. For example, $267^5$ means 267 multiplied by itself five times. When we divide terms with the same base, like we have here, there's a neat little rule we can use to make things much simpler. This rule is the quotient of powers rule, and it's the key to unlocking this problem. Let’s delve deeper into this rule and see how it applies to our specific expression. Imagine you're faced with a massive calculation, but you know there's a shortcut. That's what the quotient of powers rule is for exponents. It states that when you divide two exponents with the same base, you simply subtract the powers. Mathematically, this is expressed as: $\frac{a^m}{a^n} = a^{m-n}$. This rule is a lifesaver because it turns a division problem into a subtraction problem, which is much easier to handle. Now, let's see how this applies to our problem. We have $\frac{267^5}{267^2}$. According to the quotient of powers rule, we should subtract the exponents. This means we subtract 2 from 5, which gives us 3. So, the simplified expression will have a power of 3. It's like magic, but it's actually just math! Understanding this rule is one thing, but applying it correctly is another. Let's make sure we know exactly how to use it in this context.

Applying the Quotient of Powers Rule

Now that we've armed ourselves with the quotient of powers rule, let's put it into action. We have the expression $\frac{267^5}{267^2}$. As we discussed, the rule tells us to subtract the exponents when dividing terms with the same base. So, we'll subtract the exponent in the denominator (2) from the exponent in the numerator (5). This gives us $5 - 2 = 3$. Therefore, the simplified exponent is 3. This means our base, 267, will be raised to the power of 3. It's like we're peeling away the layers of the problem to reveal the simpler form underneath. Now, let's write this out: $267^3$. This is the simplified form of the original expression. But how do we know for sure that this is the right answer? Let's think about what the exponents actually mean. $267^5$ means 267 multiplied by itself five times, and $267^2$ means 267 multiplied by itself twice. When we divide, we're essentially canceling out two of the 267s from the numerator, leaving us with three 267s multiplied together, which is exactly what $267^3$ means. Understanding this concept helps solidify the rule in our minds and makes it easier to apply in future problems. This is a crucial step in simplifying the expression. We've taken the original complex fraction and transformed it into a much more manageable form. Now, let's match our simplified expression with the options provided in the question to find the correct answer. Remember, math is not just about getting to the answer; it's about understanding the process and the logic behind it. As we move to the next step, let’s keep this in mind and confidently select the correct option.

Matching the Simplified Expression with the Options

Alright, we've successfully simplified the expression $\frac{267^5}{267^2}$ to $267^3$. Now, the next step is to compare our simplified expression with the given options and identify the one that matches. This is like the final piece of the puzzle falling into place. We have the following options to choose from:

A. $\frac{1}{267^3}$ B. $267^3$ C. $267^7$ D. $267^{10}$

Looking at these options, it's pretty clear that option B, $267^3$, is the one that perfectly matches our simplified expression. Options A, C, and D all have different exponents or forms, so they can't be the correct answer. Option A, $\frac{1}{267^3}$, represents the reciprocal of $267^3$, which is not what we're looking for. Options C and D have exponents of 7 and 10, respectively, which are results of different operations (like adding exponents when multiplying terms with the same base, which isn't what we did here). So, with confidence, we can say that option B is the correct answer. It’s always a good idea to double-check your work, even when you’re pretty sure you’ve got it right. In this case, we can mentally retrace our steps: we used the quotient of powers rule, subtracted the exponents, and arrived at $267^3$. This matches option B. This process of matching the simplified expression with the options is a critical skill in test-taking and problem-solving. It ensures that you not only understand the concept but can also apply it to find the correct answer in a multiple-choice format. Now that we've nailed the answer, let’s recap the entire process and reinforce what we’ve learned.

Conclusion: The Correct Answer and Key Takeaways

We've successfully navigated through the problem and found the expression equivalent to $\frac{267^5}{267^2}$. After applying the quotient of powers rule, we simplified the expression to $267^3$, which corresponds to option B. So, the correct answer is B. $267^3$. But the journey to the answer is just as important as the answer itself. Let's recap the key steps we took:

1. Understanding the Problem: We recognized that we needed to simplify an expression involving exponents and division.
2. Applying the Quotient of Powers Rule: We used the rule $\frac{a^m}{a^n} = a^{m-n}$ to subtract the exponents.
3. Simplifying the Expression: We subtracted the exponents (5 - 2 = 3) to get $267^3$.
4. Matching with Options: We compared our simplified expression with the given options and identified the correct one.

This process highlights the importance of understanding the rules of exponents and knowing how to apply them. It also shows the value of breaking down a problem into smaller, manageable steps. By understanding each step, you not only arrive at the correct answer but also build a solid foundation for tackling more complex problems in the future. Remember, practice makes perfect. The more you work with exponents and apply these rules, the more confident you'll become. Exponents are a fundamental concept in mathematics, and mastering them will help you in various areas, from algebra to calculus. So, keep practicing, keep exploring, and keep learning! Before we wrap up, it’s worth noting that resources like Khan Academy's Exponents and Polynomials Section can provide further practice and deeper understanding of exponents. This external resource is excellent for continued learning and reinforcement of the concepts we've covered today. Keep up the great work, and you'll be simplifying exponents like a pro in no time!