Simplifying The Expression 2^4 * 5^6(6^6 - 3^8) / (11 * 5^7 * 2^4) A Step-by-Step Guide

Hey guys! Today, we're diving into a cool math problem that involves simplifying a complex expression. Don't worry, it's not as scary as it looks! We'll break it down step by step so it’s super easy to follow. Our mission, should we choose to accept it (and we do!), is to simplify the expression: 2⁴ . 5⁶(6⁶ - 3⁸) / (11 . 5⁷ . 2⁴). Buckle up, grab your calculators (or not, we might not even need them!), and let’s get started!

Breaking Down the Expression

First up, let’s rewrite the expression to get a clearer view of what we're dealing with. Our expression is 2⁴ . 5⁶(6⁶ - 3⁸) / (11 . 5⁷ . 2⁴). When you first look at an expression like this, it might seem overwhelming, but the key is to take it one step at a time. Remember, math problems like this are puzzles, and each piece has its place. So, let’s start by focusing on the different parts of the expression: the powers of 2, 5, and the terms inside the parenthesis. This will make it easier for us to see what we can simplify and how we can rearrange things to make the calculation smoother.

Initial Assessment

Okay, let's break this down bit by bit. We have terms involving exponents, multiplication, subtraction, and division. Phew! Sounds like a lot, but remember, we’re in this together. The first thing I notice is that we have 2⁴ in both the numerator and the denominator. That's a win! It means we can cancel those out right away. Next, we see 5⁶ in the numerator and 5⁷ in the denominator. These are also good candidates for simplification. The subtraction inside the parenthesis, 6⁶ - 3⁸, looks a bit trickier, but we'll tackle it. Our main goal is to simplify each part as much as possible before we start combining things. This approach will prevent us from getting bogged down in large numbers and will make the entire process much more manageable. Trust me; it's like decluttering your room – the more you organize, the easier it is to find what you need! So, let’s roll up our sleeves and start simplifying. This is where the real fun begins!

Simplifying Common Terms

Alright, let's get into the nitty-gritty! The beauty of this expression is that we can simplify some parts right off the bat. We’ve identified that 2⁴ appears in both the numerator and the denominator. Guess what that means? We can cancel them out! It’s like they were meant to be together and then disappear, making our expression a bit cleaner. When you see common factors like this, it's super important to take advantage of them. Think of it as free simplification – you're making the problem easier without really doing much work. It’s all about spotting those opportunities and grabbing them with both hands. Now, let’s look at the powers of 5. We have 5⁶ in the numerator and 5⁷ in the denominator. Remember the rule of exponents? When you divide terms with the same base, you subtract the exponents. So, 5⁶ / 5⁷ becomes 5^(6-7) which simplifies to 5^(-1), or 1/5. See how easy that was? By taking these initial steps to simplify the common terms, we've already made the expression significantly less intimidating. The key is to always look for these opportunities. They are like little shortcuts that help you navigate through complex problems more efficiently. Now, our expression is starting to look a lot friendlier, and we're ready to move on to the next part: simplifying the expression inside the parenthesis. Let's keep the momentum going, guys!

Tackling the Parenthesis: 6⁶ - 3⁸

Now, let’s zoom in on the part that might seem a bit intimidating at first glance: the expression inside the parentheses, 6⁶ - 3⁸. Don't worry, we've got this! The key here is to recognize that we can express both 6 and 3 as powers of smaller numbers. This will help us simplify the subtraction. Remember, math is all about finding patterns and using them to our advantage. First, let's rewrite 6⁶. We know that 6 is the same as 2 times 3, so we can rewrite 6⁶ as (2 * 3)⁶. Using the exponent rule (ab)^n = a^n * b^n, this becomes 2⁶ * 3⁶. Now, let's think about 3⁸. We already have a term with a power of 3, so this is a good sign. Our goal is to see if we can manipulate these terms so we can potentially factor something out or simplify the expression further. By breaking down the numbers into their prime factors, we are uncovering the hidden structure of the expression. It’s like looking under the hood of a car to see how the engine works. Once we understand the individual components, it becomes much easier to understand how they interact with each other. So, let's keep going! We've got 6⁶ expressed in terms of 2 and 3, and we have 3⁸. Now, the next step is to try and find a way to relate these terms to each other so we can simplify the subtraction. This might involve some clever manipulation, but we're up for the challenge!

Converting to Common Base

Okay, so we've got 6⁶ expressed as 2⁶ * 3⁶, and we have 3⁸. The next step is to try and see if we can rewrite these terms in a way that makes the subtraction easier. What do you guys think? Any ideas? Well, let’s think about this. We have powers of 3 in both terms, so maybe we can try to factor something out. To do this, we want to find the highest power of 3 that we can factor out of both terms. We have 3⁶ in the first term and 3⁸ in the second term. The highest power of 3 that divides both of these is 3⁶. So, let's try factoring that out. This is where things start to get really interesting. When we factor out 3⁶, we're essentially dividing both terms by 3⁶ and seeing what's left. This will help us simplify the expression and potentially make it much easier to calculate. Factoring is like finding the common thread that ties different pieces of a math problem together. It's a powerful technique that can transform a complex expression into something much more manageable. So, let's go ahead and factor out 3⁶ from our expression inside the parenthesis. This should give us a clearer picture of what we're dealing with and bring us one step closer to simplifying the whole thing. Remember, the key is to keep breaking down the problem into smaller, more manageable parts. We're doing great so far!

Factoring 3⁶

Alright, let's put our factoring skills to the test! We're focusing on the expression 2⁶ * 3⁶ - 3⁸ inside the parentheses. As we discussed, we want to factor out 3⁶. So, what does that look like? When we factor 3⁶ out of 2⁶ * 3⁶, we're left with 2⁶. Makes sense, right? Now, when we factor 3⁶ out of 3⁸, we need to think about what 3⁸ divided by 3⁶ is. Using our exponent rules, 3⁸ / 3⁶ = 3^(8-6) = 3². So, after factoring, our expression inside the parenthesis becomes 3⁶(2⁶ - 3²). See how much simpler that looks? We've transformed a potentially messy subtraction into a much cleaner form. This is the magic of factoring – it allows us to rewrite expressions in ways that make them easier to work with. Now we have a product instead of a difference, which is often easier to handle. And look, we've replaced those big exponents with smaller ones. We have 2⁶ and 3², which are much less intimidating than 6⁶ and 3⁸. The key takeaway here is that factoring is a powerful tool in your math arsenal. It allows you to simplify expressions, solve equations, and generally make your life easier when dealing with complex problems. So, now that we've successfully factored out 3⁶, let’s keep going. We're making great progress, and the simplified form is starting to take shape!

Simplifying Further

Okay, we've factored out 3⁶ and now we have 3⁶(2⁶ - 3²). Let's keep simplifying! The next step is to evaluate the expression inside the parenthesis: 2⁶ - 3². We need to calculate 2⁶ and 3². These are much smaller numbers now, which is great. 2⁶ means 2 multiplied by itself six times, which is 2 * 2 * 2 * 2 * 2 * 2 = 64. Easy peasy! And 3² is simply 3 * 3, which equals 9. So, now we have 64 - 9. What’s that equal? It's 55! Now our expression inside the parenthesis has been simplified to a single number. We’ve gone from 6⁶ - 3⁸ to 3⁶(55). That's a huge improvement! This step highlights the importance of carrying out the operations in the correct order. We tackled the exponents first, then the subtraction. This is crucial to getting the correct answer. By breaking down the problem into manageable steps, we've avoided making mistakes and have arrived at a much simpler form. The key is to take your time and make sure you're following the correct mathematical principles. Now that we've simplified the parenthesis, we're ready to put everything back together and see where we stand with the entire expression. We're on the home stretch now, guys! Let’s keep going and nail this problem.

Putting It All Together

Alright, we’ve done some serious simplifying, and it’s time to bring everything together. Let’s recap where we are. We started with the expression 2⁴ . 5⁶(6⁶ - 3⁸) / (11 . 5⁷ . 2⁴). We canceled out the 2⁴ terms, simplified the powers of 5, and transformed the expression inside the parenthesis from 6⁶ - 3⁸ to 3⁶(55). Now, let's rewrite the entire expression with these simplifications. After canceling 2⁴, we were left with 5⁶(6⁶ - 3⁸) / (11 . 5⁷). Then, we simplified 5⁶ / 5⁷ to 1/5. And we simplified the parenthesis to 3⁶(55). So, our expression now looks like this: (1/5) * 3⁶(55) / 11. See how much cleaner that is compared to where we started? This is the power of simplification! By tackling each part of the problem step by step, we've transformed a complex-looking expression into something much more manageable. Now, let's take a moment to appreciate how far we've come. We've used exponent rules, factoring, and basic arithmetic to break down this problem. This shows how all the different concepts in math are interconnected and how mastering the basics can help you solve more complex problems. Now, we’re just a few steps away from the final answer. Let’s keep pushing and finish strong! We're in the final stretch, guys!

Final Simplification

Okay, so we're at the point where our expression looks like (1/5) * 3⁶(55) / 11. Let's finish this off! The next thing I notice is that we have 55 in the numerator and 11 in the denominator. Ding ding ding! These have a common factor! 55 is 5 times 11, so we can simplify 55 / 11 to 5. This makes our expression even simpler. Now we have (1/5) * 3⁶ * 5. What’s next? We have a 1/5 and a 5. These are reciprocals, so they cancel each other out! (1/5) * 5 = 1. It’s like they were destined to be together and vanish, leaving us with just 3⁶. So, our expression has been simplified all the way down to 3⁶. Wow! That’s quite a journey from where we started. But we're not quite done yet. We need to calculate 3⁶. This means 3 multiplied by itself six times: 3 * 3 * 3 * 3 * 3 * 3. This is 9 * 9 * 9, which is 81 * 9, which equals 729. So, after all that simplification, our final answer is 729! How awesome is that? We took a complicated-looking expression and, step by step, we simplified it down to a single number. This is what math is all about: taking something complex and breaking it down into manageable pieces. We've shown that with the right approach and a little bit of know-how, even the most intimidating problems can be solved. Give yourselves a pat on the back, guys! You’ve earned it!

Final Answer

So, after all the simplifying and calculating, we've arrived at our final answer. The expression 2⁴ . 5⁶(6⁶ - 3⁸) / (11 . 5⁷ . 2⁴) simplifies to 729. Boom! We did it! This whole process really shows the power of breaking down complex problems into smaller, more manageable steps. We used several key mathematical concepts along the way, including exponent rules, factoring, and basic arithmetic. Each step was like a piece of the puzzle, and by putting them all together, we revealed the final picture. It’s like we were math detectives, uncovering clues and solving the mystery. The journey from the original expression to our final answer might have seemed daunting at first, but by staying organized and methodical, we were able to navigate through it successfully. Remember, math isn't about memorizing formulas; it's about understanding the underlying principles and applying them creatively. We saw that in action today! We took a seemingly intimidating problem and transformed it into something simple and elegant. So, the next time you encounter a complex math problem, don't panic! Take a deep breath, break it down, and remember the techniques we used today. You've got this! And that’s a wrap, folks! I hope you enjoyed this math adventure as much as I did. Keep practicing, keep exploring, and most importantly, keep having fun with math! Until next time!