Expressions Equivalent To 12^4 / 6^4 Explained

by Scholario Team 47 views

Hey guys! Today, we're diving into a fun math problem that involves exponents and fractions. We're going to figure out which expressions are equivalent to 12464\frac{12^4}{6^4}. This is a fantastic way to brush up on our exponent rules and simplify some tricky-looking calculations. So, let's get started and break it down step by step!

Understanding the Problem

First, let's make sure we understand what the question is asking. We have the expression 12464\frac{12^4}{6^4}, and we need to find other expressions that have the same value. This means we're looking for equivalent expressions. Remember, equivalent expressions might look different, but they all boil down to the same numerical result. This is a classic problem that tests your understanding of exponent rules and simplification techniques. Think of it like this: we're trying to find different paths that lead to the same destination. So, how do we approach this? Let's explore the world of exponents and see what we can uncover!

Key Takeaway: We're looking for expressions that are mathematically equal to 12464\frac{12^4}{6^4}, even if they look different on the surface.

Breaking Down the Expression

One of the best ways to tackle problems like this is to break them down into smaller, more manageable parts. So, let’s start by rewriting the given expression. We have 12464\frac{12^4}{6^4}. Notice anything interesting? Both the numerator (12412^4) and the denominator (646^4) have the same exponent, which is 4. This is a huge clue! It means we can use one of our handy exponent rules to simplify things. Do you remember the rule that says (ab)n=anbn(\frac{a}{b})^n = \frac{a^n}{b^n}? Well, we can use this rule in reverse! We can rewrite 12464\frac{12^4}{6^4} as (126)4(\frac{12}{6})^4. See how much simpler that looks already? We've taken a potentially complex fraction with large exponents and turned it into something much easier to handle. By recognizing this key property, we're setting ourselves up for success in the rest of the problem.

Key Takeaway: Rewriting 12464\frac{12^4}{6^4} as (126)4(\frac{12}{6})^4 simplifies the problem using exponent rules.

Simplifying the Base

Now that we've rewritten our expression as (126)4(\frac{12}{6})^4, let's focus on the fraction inside the parentheses: 126\frac{12}{6}. This is a simple fraction, and we can simplify it easily. What is 12 divided by 6? It's 2, of course! So, we can replace 126\frac{12}{6} with 2. This gives us (2)4(2)^4, which is just 242^4. Wow, we've made some serious progress! We started with a complex-looking expression and, through the magic of simplification, we've arrived at 242^4. This is a much more manageable form, and it makes it easier to compare with the answer choices. Remember, the goal is to find expressions that are equivalent to the original, and we've just found a significantly simpler equivalent expression. This step highlights the power of simplifying fractions within exponents to make the overall expression more understandable and workable. So, let's hold onto this simplified form and move forward!

Key Takeaway: Simplifying the fraction 126\frac{12}{6} to 2 results in the expression 242^4.

Evaluating the Options

Okay, we've simplified the original expression 12464\frac{12^4}{6^4} to 242^4. Now it's time to put on our detective hats and see which of the given options are also equal to 242^4. This is where we'll really put our understanding of exponents to the test. We'll go through each option one by one, carefully evaluating it and comparing it to our simplified form. This is a crucial step because it's not enough to just simplify the original expression; we need to make sure we can recognize its equivalent forms in different guises. So, let’s dive into those options and see what we find!

Option A: 242^4

Let's start with the first option, which is 242^4. This one is pretty straightforward, guys! We've already simplified our original expression to 242^4, so this option is clearly equivalent. It's like finding the treasure right at the beginning of the map! But don't let that fool you; we still need to check the other options to make sure we haven't missed anything. Sometimes there can be multiple correct answers, so it's always good to be thorough. This option serves as a great confirmation that we're on the right track and that our simplification process was accurate. It's a solid starting point for evaluating the remaining choices.

Key Takeaway: 242^4 is equivalent to our simplified expression, so it's a correct answer.

Option B: 2^2 ullet 2^2

Now let's tackle Option B, which is 2^2 ullet 2^2. At first glance, it might not be immediately obvious if this is equivalent to 242^4. But remember your exponent rules! When we multiply terms with the same base, we add the exponents. So, 2^2 ullet 2^2 is the same as 22+22^{2+2}, which simplifies to 242^4. Bingo! This option is also equivalent to our simplified expression. See how important it is to remember those exponent rules? They allow us to manipulate expressions and see their hidden connections. This option shows us that there can be different ways to represent the same value, and understanding exponent rules is key to unlocking those representations. We're building up a good list of equivalent expressions here!

Key Takeaway: Using the exponent rule for multiplication, 2^2 ullet 2^2 simplifies to 242^4, making it another correct answer.

Option C: 124\frac{1}{2^4}

Let's move on to Option C: 124\frac{1}{2^4}. This one looks a bit different from what we've seen so far. We have a fraction with 1 in the numerator and 242^4 in the denominator. To determine if this is equivalent to 242^4, we need to think about negative exponents. Remember, a term with a negative exponent can be rewritten as its reciprocal with a positive exponent. In other words, a−n=1ana^{-n} = \frac{1}{a^n}. So, 124\frac{1}{2^4} can be rewritten as 2−42^{-4}. Is 2−42^{-4} the same as 242^4? Absolutely not! The negative exponent indicates a reciprocal, so this option represents the inverse of 242^4. This option is a great example of how a small change, like a negative sign in the exponent, can drastically change the value of an expression. We can confidently rule this one out.

Key Takeaway: 124\frac{1}{2^4} is equivalent to 2−42^{-4}, which is not equal to 242^4, so this option is incorrect.

Option D: 2

Finally, let's consider Option D: 2. This is the simplest option of the bunch, but don't let that simplicity fool you. We need to be absolutely sure whether or not it's equivalent to 242^4. We know that 242^4 means 2 multiplied by itself four times: 2 ullet 2 ullet 2 ullet 2. That's equal to 16. Is 2 the same as 16? Nope! This option is a clear mismatch. It's a good reminder that even simple-looking expressions can be traps if we don't carefully evaluate them. This option reinforces the importance of understanding the fundamental meaning of exponents and how they affect the value of an expression. We've thoroughly examined all the options, and we're ready to declare our final answer!

Key Takeaway: 2 is not equal to 242^4, so this option is incorrect.

The Solution

Alright, guys! We've done it! We've successfully navigated through the maze of exponents and fractions, and we're ready to reveal the solution. After carefully simplifying the original expression 12464\frac{12^4}{6^4} and evaluating each option, we found that two options are equivalent:

  • A. 242^4
  • B. 2^2 ullet 2^2

These are the expressions that hold the same mathematical value as our starting point. We broke down the problem step by step, using exponent rules and simplification techniques to arrive at our answer. This problem highlights the importance of not just memorizing rules, but understanding how to apply them in different situations. By simplifying the original expression and then comparing it to the options, we were able to confidently identify the correct answers. Math can be like a puzzle, and each step we take brings us closer to the satisfying solution!

Final Answer: The expressions equivalent to 12464\frac{12^4}{6^4} are 242^4 and 2^2 ullet 2^2.

Final Thoughts

So there you have it! We've successfully solved this problem and learned a few valuable lessons along the way. Remember, when you encounter problems involving exponents and fractions, don't be intimidated. Break them down into smaller parts, look for opportunities to simplify, and remember your exponent rules. Math is all about building on your knowledge and applying it in creative ways. Keep practicing, keep exploring, and you'll become a math whiz in no time! And most importantly, don't forget to have fun with it! Math can be challenging, but it's also incredibly rewarding when you crack the code and find the solution. Keep up the great work, guys! You've got this!