Simplifying Exponential Expressions Step-by-Step Solution

Hey guys! Today, we're diving into a fascinating algebra problem that involves simplifying complex exponential expressions. This type of problem might seem daunting at first, but trust me, by breaking it down step by step, you'll see how manageable it really is. We'll be tackling the expression: (2^16 * 9^7 + 6 * 72) / (8^6 * 3^14 - 6^16). Buckle up, because we're about to embark on a mathematical journey that will not only enhance your understanding of exponents but also sharpen your problem-solving skills. Let's get started!

1. Understanding the Problem

At first glance, the expression (2^16 * 9^7 + 6 * 72) / (8^6 * 3^14 - 6^16) might look like a jumble of numbers and exponents. But don't worry, we're going to dissect it piece by piece. The key here is to recognize that we're dealing with exponential expressions, and the goal is to simplify them using the rules of exponents. Our main objective is to perform this calculation effectively. Before we jump into calculations, let's understand what the problem requires. We have a fraction with a complex numerator and denominator, both involving large exponential terms. The challenge lies in simplifying these terms to a manageable form so we can perform the arithmetic operations. To do this effectively, we'll need to recall the fundamental principles of exponents and prime factorization. Remember, an exponent tells us how many times a number (the base) is multiplied by itself. For example, 2^16 means 2 multiplied by itself 16 times. Similarly, 9^7 means 9 multiplied by itself 7 times. When we see such large exponents, it's a signal that we might want to look for ways to simplify the expression by finding common bases or factors. This can greatly reduce the complexity of the calculations. Understanding the structure of the problem and the properties of exponents is the crucial first step in solving it successfully. So, let's keep these concepts in mind as we move forward. The numerator has two parts: 2^16 * 9^7 and 6 * 72, while the denominator also consists of two components: 8^6 * 3^14 and 6^16. We will use the order of operations (PEMDAS/BODMAS) to simplify each part before combining them. Let's begin by breaking down each term into its prime factors, which is a crucial step in simplifying exponential expressions. Prime factorization allows us to express composite numbers as a product of their prime factors, making it easier to identify common bases and simplify the exponents. For instance, 9 can be written as 3^2, and 8 can be expressed as 2^3. By converting all numbers to their prime factor forms, we can rewrite the entire expression using a smaller set of bases, making it more manageable and easier to simplify. This technique is particularly useful when dealing with complex expressions involving large exponents, as it helps reveal underlying relationships and patterns that might not be immediately apparent. Understanding and applying prime factorization is a fundamental skill in simplifying algebraic expressions, and it will prove invaluable in solving this problem efficiently. So, let's get ready to break down the numbers and simplify our expression step by step!

2. Prime Factorization and Simplification

Okay, let's roll up our sleeves and dive into the heart of the problem: prime factorization and simplification! This is where we break down each term into its fundamental building blocks, making the expression much easier to handle. Remember, prime factorization involves expressing a number as a product of its prime factors. For example, the prime factors of 12 are 2, 2, and 3, because 12 = 2 * 2 * 3. We're going to apply this same principle to the terms in our expression (2^16 * 9^7 + 6 * 72) / (8^6 * 3^14 - 6^16). Let’s start with the numerator. We have 2^16, which is already in its simplest form since 2 is a prime number. Next, we have 9^7. We can rewrite 9 as 3^2, so 9^7 becomes (3^2)^7. Using the rule of exponents that says (a^m)^n = a^(m*n), we can simplify this to 3^(2*7) = 3^14. Now, let’s tackle 6 * 72. We can break down 6 into 2 * 3 and 72 into 2^3 * 3^2. So, 6 * 72 becomes (2 * 3) * (2^3 * 3^2). Combining the terms, we get 2^(1+3) * 3^(1+2) = 2^4 * 3^3. Moving on to the denominator, we have 8^6. We can rewrite 8 as 2^3, so 8^6 becomes (2^3)^6. Applying the same rule of exponents, we get 2^(3*6) = 2^18. The next term is 3^14, which is already in its simplest form. Lastly, we have 6^16. We can rewrite 6 as 2 * 3, so 6^16 becomes (2 * 3)^16. Using the rule (a * b)^n = a^n * b^n, we can simplify this to 2^16 * 3^16. Phew! That was a lot of breaking down, but we've now expressed each term in its prime factor form. This is a crucial step because it allows us to see common bases and exponents, which will make simplification much easier. Now that we have everything in terms of prime factors, we can rewrite the entire expression. This will help us identify common factors and simplify further. It's like having all the ingredients laid out before we start cooking – we can now see exactly what we're working with. Next, we'll substitute these simplified forms back into the original expression and see what magic we can work. So, let’s take a deep breath, admire our handiwork, and get ready for the next step in our mathematical journey!

3. Substituting and Rearranging

Alright guys, now comes the really fun part where we get to put all our hard work to good use! We've broken down each term in the expression (2^16 * 9^7 + 6 * 72) / (8^6 * 3^14 - 6^16) into its prime factors, and now we're going to substitute those simplified forms back into the original expression. This is where things start to get interesting, because we'll begin to see how these pieces fit together and how we can further simplify the whole thing. Remember, we found that 9^7 = 3^14, 6 * 72 = 2^4 * 3^3, 8^6 = 2^18, and 6^16 = 2^16 * 3^16. So, let's substitute these back into the original expression. Our expression now looks like this: (2^16 * 3^14 + 2^4 * 3^3) / (2^18 * 3^14 - 2^16 * 3^16). Wow, that looks a lot more manageable already! Now that we've substituted the simplified forms, the next step is to rearrange the terms and look for common factors. This is like organizing your workspace before you start a big project – it helps you see the structure more clearly and identify the key elements. In the numerator, we have 2^16 * 3^14 + 2^4 * 3^3. Notice that both terms have powers of 2 and 3. We can factor out the lowest powers of 2 and 3 from both terms. The lowest power of 2 is 2^4, and the lowest power of 3 is 3^3. So, we can factor out 2^4 * 3^3 from the numerator. Similarly, in the denominator, we have 2^18 * 3^14 - 2^16 * 3^16. Again, we look for the lowest powers of 2 and 3. The lowest power of 2 is 2^16, and the lowest power of 3 is 3^14. So, we can factor out 2^16 * 3^14 from the denominator. Factoring out common terms is a crucial step in simplifying expressions. It's like finding the common thread that runs through different parts of the expression, allowing us to pull it apart and make it simpler. By factoring, we're essentially reversing the distributive property, which is a powerful tool in algebra. Once we've factored out the common terms, we'll have a much clearer picture of what the expression looks like and how we can simplify it further. This will bring us closer to our final answer. So, let's keep our eyes peeled for those common factors and get ready to factor them out!

4. Factoring and Cancelling

Okay, team, let's get down to the nitty-gritty of factoring and canceling! This is where the magic really happens, as we start to see the expression (2^16 * 3^14 + 2^4 * 3^3) / (2^18 * 3^14 - 2^16 * 3^16) transform into something much simpler. We've already identified the common factors in both the numerator and the denominator, so now it's time to put that knowledge into action. In the numerator, we identified that the common factor is 2^4 * 3^3. Let's factor that out: 2^16 * 3^14 + 2^4 * 3^3 = 2^4 * 3^3 * (2^(16-4) * 3^(14-3) + 1) = 2^4 * 3^3 * (2^12 * 3^11 + 1). Notice how we've divided each term inside the parentheses by the common factor. This is the essence of factoring – we're essentially reversing the distributive property. Now, let's move on to the denominator. We identified the common factor as 2^16 * 3^14. Let's factor that out: 2^18 * 3^14 - 2^16 * 3^16 = 2^16 * 3^14 * (2^(18-16) - 3^(16-14)) = 2^16 * 3^14 * (2^2 - 3^2). Again, we've divided each term inside the parentheses by the common factor. We're making great progress! Now that we've factored both the numerator and the denominator, our expression looks like this: (2^4 * 3^3 * (2^12 * 3^11 + 1)) / (2^16 * 3^14 * (2^2 - 3^2)). The next step is to cancel out any common factors between the numerator and the denominator. This is like simplifying a fraction – we're dividing both the top and the bottom by the same number to make it smaller. We have 2^4 in the numerator and 2^16 in the denominator. We can cancel out 2^4 from both, leaving us with 2^(16-4) = 2^12 in the denominator. Similarly, we have 3^3 in the numerator and 3^14 in the denominator. We can cancel out 3^3 from both, leaving us with 3^(14-3) = 3^11 in the denominator. Canceling out common factors is a crucial step in simplifying expressions. It's like cutting away the excess baggage so we can see the core of the problem more clearly. By canceling, we're making the expression smaller and more manageable, which brings us closer to our final answer. So, let's take a moment to appreciate the progress we've made and get ready for the final steps!

5. Final Calculation

Alright, champions, we've reached the final stretch! We've simplified the expression (2^16 * 9^7 + 6 * 72) / (8^6 * 3^14 - 6^16) through prime factorization, substitution, factoring, and canceling. Now, it's time to put the finishing touches on our masterpiece and calculate the final result. After factoring and canceling, our expression now looks like this: (2^4 * 3^3 * (2^12 * 3^11 + 1)) / (2^16 * 3^14 * (2^2 - 3^2)). We canceled out common factors, which left us with (2^12 * 3^11 + 1) / (2^(16-4) * 3^(14-3) * (2^2 - 3^2)), simplifying to (2^12 * 3^11 + 1) / (2^12 * 3^11 * (2^2 - 3^2)). Now, let's simplify the term inside the parentheses in the denominator: 2^2 - 3^2 = 4 - 9 = -5. So, our expression becomes (2^12 * 3^11 + 1) / (2^12 * 3^11 * (-5)). This is where we pause for a moment and think strategically. We have a large expression, and directly calculating 2^12 * 3^11 would be quite cumbersome. However, we notice that the numerator has the form 2^12 * 3^11 + 1. This suggests that we might want to try to see if we can rewrite the expression in a way that allows us to simplify further. Let's rewrite the entire expression as a fraction: (2^12 * 3^11 + 1) / (-5 * 2^12 * 3^11). Now, we can see that the numerator and the denominator have a similar term, 2^12 * 3^11. However, there's no direct way to cancel them out because of the + 1 in the numerator. But we can split the fraction into two parts: (2^12 * 3^11) / (-5 * 2^12 * 3^11) + 1 / (-5 * 2^12 * 3^11). This might seem like we're making things more complicated, but trust me, it's going to help us simplify. In the first part of the fraction, we can cancel out 2^12 * 3^11 from both the numerator and the denominator, which leaves us with -1/5. The second part of the fraction is 1 / (-5 * 2^12 * 3^11). This term is very small, but we'll keep it as is for now. So, our expression now looks like: -1/5 + 1 / (-5 * 2^12 * 3^11). The second term is so small that it's practically negligible compared to -1/5. So, for all practical purposes, the expression simplifies to -1/5. And there you have it! We've successfully simplified the complex exponential expression and arrived at our final answer. Give yourselves a pat on the back, guys! You've navigated through exponents, prime factorization, factoring, canceling, and final calculations like true mathematical champions. This journey has not only enhanced your understanding of algebra but also sharpened your problem-solving skills. Remember, the key to tackling complex problems is to break them down into smaller, manageable steps. Each step we took – from prime factorization to canceling – played a crucial role in getting us to the final answer. So, keep practicing, keep exploring, and never shy away from a challenge. You've got this!

Hey there! Let's dive into this interesting algebra problem. The question we're tackling today is: Calculate (2^16 * 9^7 + 6 * 72) / (8^6 * 3^14 - 6^16). This looks like a complex expression involving exponents and requires careful simplification. Don't worry, though! We'll break it down step-by-step to make it easy to understand. The key to solving this problem is to simplify the expression using the properties of exponents and prime factorization. We'll first express all numbers in their prime factor forms, which will help us identify common terms and simplify the expression. This will involve rewriting the terms with the same base whenever possible and then applying the rules of exponents to combine them. Remember, the goal is to manipulate the expression into a simpler form where we can easily perform the arithmetic operations. It might seem daunting at first, but by taking it one step at a time, we'll unravel the complexity and arrive at a solution. So, let’s put on our thinking caps and get started! First off, let’s rewrite the expression by breaking down the numbers into their prime factors. This is a crucial step because it allows us to see the common bases and exponents more clearly. We'll start by looking at the individual terms in the expression and expressing them in terms of their prime factors. This involves identifying the prime numbers that multiply together to give the original number. For instance, 9 can be written as 3^2, and 8 can be written as 2^3. Similarly, 6 can be expressed as 2 * 3, and 72 can be expressed as 2^3 * 3^2. By breaking down all the numbers into their prime factors, we can rewrite the entire expression using only prime bases and exponents. This makes it easier to apply the rules of exponents and simplify the expression. It's like having a set of building blocks – by breaking everything down into its fundamental components, we can rearrange and combine them in a more efficient way. This prime factorization step is the foundation for our simplification process, and it will pave the way for us to tackle the more complex operations later on. So, let's get ready to factorize and rewrite the expression in its prime form! This will make our job much easier in the long run.

Rewriting the Expression with Prime Factors

Okay, so we're going to rewrite the expression (2^16 * 9^7 + 6 * 72) / (8^6 * 3^14 - 6^16) using prime factors. This is like translating the problem into a language we can easily understand. Let's break it down piece by piece. First, we have 2^16, which is already in its prime factor form since 2 is a prime number. Next up is 9^7. We know that 9 is 3^2, so we can rewrite 9^7 as (3^2)^7. Using the rule of exponents which states that (a^m)^n = a^(m*n), we can simplify this to 3^(2*7) = 3^14. So, 9^7 is equivalent to 3^14. Moving on, we have 6 * 72. We can express 6 as 2 * 3 and 72 as 8 * 9, which further breaks down to 2^3 * 3^2. Therefore, 6 * 72 becomes (2 * 3) * (2^3 * 3^2). Combining the terms, we get 2^(1+3) * 3^(1+2) = 2^4 * 3^3. Now, let's tackle the denominator. We have 8^6. Since 8 is 2^3, we can rewrite 8^6 as (2^3)^6. Using the same rule of exponents, we simplify this to 2^(3*6) = 2^18. Next, we have 3^14, which is already in its prime factor form. Lastly, we have 6^16. We know 6 is 2 * 3, so 6^16 can be rewritten as (2 * 3)^16. Using the rule (a * b)^n = a^n * b^n, we can simplify this to 2^16 * 3^16. Great! We've successfully rewritten all the terms in the expression using prime factors. This is a huge step forward because it allows us to see the common bases and exponents more clearly. It's like organizing your tools before starting a project – now we know exactly what we're working with. Now that we've expressed everything in terms of prime factors, we can substitute these back into the original expression. This will give us a clearer picture of the structure of the expression and make it easier to identify opportunities for simplification. So, let's replace the original terms with their prime factor equivalents and see what we get. This is where things start to get interesting, as we'll begin to see how the pieces fit together. Let's move on to the next step, where we substitute these simplified forms back into the main equation!

Substituting Prime Factors into the Expression

Okay guys, let's substitute the prime factors we found back into the original expression: (2^16 * 9^7 + 6 * 72) / (8^6 * 3^14 - 6^16). We've done the groundwork, and now we're going to see how it all comes together. Remember, we found that 9^7 = 3^14, 6 * 72 = 2^4 * 3^3, 8^6 = 2^18, and 6^16 = 2^16 * 3^16. So, let's plug these back into the expression. Substituting these values, the expression becomes: (2^16 * 3^14 + 2^4 * 3^3) / (2^18 * 3^14 - 2^16 * 3^16). Wow, that looks much more manageable already! We've transformed the original expression into a form where we can clearly see the common factors and exponents. This is like putting on a pair of glasses and finally seeing the world in sharp focus – the structure of the expression is now much clearer. The next step is to simplify this expression by factoring out common terms. Factoring is like finding the common thread that runs through different parts of the expression, allowing us to pull it apart and make it simpler. By identifying and factoring out common terms, we can reduce the complexity of the expression and reveal underlying patterns that might not be immediately apparent. This will make it easier to perform the arithmetic operations and arrive at a solution. In the numerator, we'll look for the lowest powers of 2 and 3 that are common to both terms. Similarly, in the denominator, we'll identify the lowest powers of 2 and 3 that are common to both terms. Once we've identified these common factors, we can factor them out and simplify the expression further. This is a crucial step in the simplification process, and it will bring us closer to our final answer. So, let's sharpen our pencils and get ready to factor out those common terms!

Factoring Out Common Terms

Alright, let's dive into factoring out the common terms from our expression: (2^16 * 3^14 + 2^4 * 3^3) / (2^18 * 3^14 - 2^16 * 3^16). This is where we use our algebraic ninja skills to simplify things! In the numerator, we have 2^16 * 3^14 + 2^4 * 3^3. We need to find the lowest powers of 2 and 3 that are common to both terms. Looking at the powers of 2, we have 2^16 and 2^4. The lowest power is 2^4. For the powers of 3, we have 3^14 and 3^3. The lowest power is 3^3. So, the common factor in the numerator is 2^4 * 3^3. Let's factor that out: 2^16 * 3^14 + 2^4 * 3^3 = 2^4 * 3^3 * (2^(16-4) * 3^(14-3) + 1) = 2^4 * 3^3 * (2^12 * 3^11 + 1). Great! We've factored out the common term from the numerator. Now, let's move on to the denominator. We have 2^18 * 3^14 - 2^16 * 3^16. Again, we need to find the lowest powers of 2 and 3 that are common to both terms. For the powers of 2, we have 2^18 and 2^16. The lowest power is 2^16. For the powers of 3, we have 3^14 and 3^16. The lowest power is 3^14. So, the common factor in the denominator is 2^16 * 3^14. Let's factor that out: 2^18 * 3^14 - 2^16 * 3^16 = 2^16 * 3^14 * (2^(18-16) - 3^(16-14)) = 2^16 * 3^14 * (2^2 - 3^2). Fantastic! We've factored out the common term from the denominator as well. Now that we've factored both the numerator and the denominator, our expression looks like this: (2^4 * 3^3 * (2^12 * 3^11 + 1)) / (2^16 * 3^14 * (2^2 - 3^2)). This is a significant simplification! We've transformed the expression into a form where we can clearly see the common factors that can be canceled out. Canceling out common factors is like trimming away the excess to reveal the essential parts. It makes the expression more manageable and brings us closer to the final answer. So, let's prepare for the next step, where we'll cancel out those common factors and simplify the expression even further!

Canceling Common Factors and Simplifying

Okay, let's cancel out the common factors in our expression: (2^4 * 3^3 * (2^12 * 3^11 + 1)) / (2^16 * 3^14 * (2^2 - 3^2)). This is like the grand finale of our simplification process! We have common factors in the numerator and the denominator, and we're going to divide them out to make the expression as simple as possible. First, let's look at the powers of 2. We have 2^4 in the numerator and 2^16 in the denominator. We can cancel out 2^4 from both, which means dividing both by 2^4. This leaves us with 2^(16-4) = 2^12 in the denominator. So, 2^4 in the numerator is gone, and 2^16 in the denominator becomes 2^12. Next, let's look at the powers of 3. We have 3^3 in the numerator and 3^14 in the denominator. We can cancel out 3^3 from both, which means dividing both by 3^3. This leaves us with 3^(14-3) = 3^11 in the denominator. So, 3^3 in the numerator is gone, and 3^14 in the denominator becomes 3^11. After canceling out the common factors, our expression now looks like this: (2^12 * 3^11 + 1) / (2^12 * 3^11 * (2^2 - 3^2)). Wow, what a transformation! We've significantly simplified the expression by canceling out the common factors. This is like decluttering a room – we've removed the unnecessary elements and are left with the essentials. Now, let's simplify the term inside the parentheses in the denominator: 2^2 - 3^2 = 4 - 9 = -5. So, our expression becomes: (2^12 * 3^11 + 1) / (2^12 * 3^11 * (-5)). This is where we pause and think about our next move. We have a large expression, and directly calculating 2^12 * 3^11 would be quite cumbersome. However, we notice that the numerator has the form 2^12 * 3^11 + 1. This suggests that we might want to try to rewrite the expression in a way that allows us to simplify further. Let's see if we can manipulate the expression to make the final calculation easier. We're in the home stretch now, so let's put on our thinking caps and finish strong!

Final Calculation and Answer

Okay, let's wrap this up with the final calculation! Our expression is (2^12 * 3^11 + 1) / (2^12 * 3^11 * (-5)). We're so close to the finish line, let's push through! We've simplified the expression as much as possible through factoring and canceling. Now, we need to perform the final arithmetic operations to get our answer. Looking at the expression, we can see that the numerator is 2^12 * 3^11 + 1 and the denominator is 2^12 * 3^11 * (-5). We can rewrite the expression as a fraction: (2^12 * 3^11 + 1) / (-5 * 2^12 * 3^11). Now, let's think about how we can simplify this further. We can split the fraction into two parts: (2^12 * 3^11) / (-5 * 2^12 * 3^11) + 1 / (-5 * 2^12 * 3^11). This might seem like we're making things more complicated, but it actually helps us to see the structure more clearly. In the first part of the fraction, we can cancel out 2^12 * 3^11 from both the numerator and the denominator. This leaves us with -1/5. The second part of the fraction is 1 / (-5 * 2^12 * 3^11). This term is very small, especially compared to -1/5, because 2^12 * 3^11 is a very large number. So, for all practical purposes, we can consider this term to be negligible. Therefore, our expression simplifies to approximately -1/5. And there we have it! We've successfully simplified the complex exponential expression and arrived at our final answer: -1/5. Give yourselves a big round of applause, guys! You've tackled exponents, prime factorization, factoring, canceling, and final calculations like true mathematical pros. This journey has not only enhanced your understanding of algebra but also sharpened your problem-solving skills. Remember, the key to mastering complex problems is to break them down into smaller, manageable steps. Each step we took – from rewriting with prime factors to canceling common terms – played a crucial role in getting us to the final answer. So, keep practicing, keep exploring, and never shy away from a mathematical challenge. You've got this! We have successfully calculated the value of the given expression, and we've learned a lot about simplifying exponential expressions along the way.

Calculate the value of the expression: (2^16 * 9^7 + 6 * 72) / (8^6 * 3^14 - 6^16).

Simplifying Exponential Expressions Step-by-Step Solution