Rewrite (4^11)(4^-8) As 4^n: A Step-by-Step Guide

Hey everyone! Today, we're diving into the fascinating world of exponential expressions, specifically focusing on how to rewrite expressions in the form of $4^n$. This is a crucial skill in mathematics, especially when dealing with algebra, calculus, and various scientific applications. Understanding how to manipulate exponents not only simplifies complex calculations but also provides a deeper insight into the nature of mathematical relationships. Let's break down the basics and then tackle a specific problem to illustrate the process. So, grab your thinking caps, and let's get started!

At its core, an exponential expression represents repeated multiplication. The expression $4^n$ means that we are multiplying the base, which is 4 in this case, by itself n times. The number n is called the exponent, or sometimes the power. For instance, $4^2$ means 4 multiplied by itself, which equals 16. Similarly, $4^3$ means 4 multiplied by itself three times, which equals 64. Understanding this fundamental concept is the key to mastering exponential expressions. But what happens when we start dealing with different exponents, like in the expression we're about to solve? That's where the laws of exponents come into play. These laws are like the secret code that unlocks the mysteries of exponents, allowing us to simplify and rewrite expressions with ease. One of the most important laws, and the one we'll use today, is the product of powers rule. This rule states that when you multiply two exponential expressions with the same base, you can add their exponents. Mathematically, this is written as $a^m * a^n = a^(m+n)$. This rule is a game-changer because it transforms a multiplication problem into a simpler addition problem in the exponent. Another crucial concept to grasp is the idea of negative exponents. A negative exponent indicates the reciprocal of the base raised to the positive version of that exponent. For example, $4^{-1}$ is the same as $\frac{1}{4}$, and $4^{-2}$ is the same as $\frac{1}{4^2}$ or $\frac{1}{16}$. Negative exponents might seem a bit strange at first, but they are incredibly useful for representing fractions and small values in a concise way. They also play a vital role in simplifying expressions and making calculations more manageable. So, with these fundamental concepts in our toolkit – the basic definition of exponential expressions, the product of powers rule, and the understanding of negative exponents – we are now well-equipped to tackle the problem at hand. Let's move on and see how we can apply these principles to rewrite the given expression in the desired form.

Applying the Product of Powers Rule

Okay, guys, let's dive into the specific problem we have: $(4^{11})(4^{-8}) = \square$. Our goal here is to rewrite this expression in the form $4^n$. This means we need to simplify the expression and figure out what value n should be. Don't worry, it's simpler than it looks! We're going to use the product of powers rule, which, as we discussed earlier, is a cornerstone of simplifying exponential expressions. This rule is our best friend when we're dealing with expressions that have the same base but different exponents. It essentially allows us to combine these expressions into a single term, making the whole thing much easier to handle. Remember, the product of powers rule states that $a^m * a^n = a^(m+n)$. In our case, the base a is 4, the exponent m is 11, and the exponent n is -8. See how nicely this fits into our rule? All we need to do is add the exponents together. This is where the magic happens. By applying the product of powers rule, we transform the multiplication problem $(4^{11})(4^{-8})$ into a simple addition problem in the exponent: $4^(11 + (-8))$. Now, it's just a matter of performing the addition. We're adding a positive number (11) and a negative number (-8). Think of it like this: you have 11 apples, and you're taking away 8 apples. How many apples do you have left? You have 3 apples! So, 11 + (-8) equals 3. Therefore, our simplified expression becomes $4^3$. We've successfully combined the two exponential terms into one, and we've determined the value of the exponent. This is a huge step! But let's not stop here. It's always a good idea to double-check our work and make sure our answer makes sense. We can do this by thinking about what the original expression means and comparing it to our simplified expression. The original expression, $(4^{11})(4^{-8})$, means we're multiplying 4 by itself 11 times and then multiplying that result by 4 raised to the power of -8. Remember, a negative exponent means we're dealing with a reciprocal. So, $4^{-8}$ is the same as $\frac{1}{4^8}$. This means we're essentially dividing by 4 eight times. When we multiply $4^{11}$ by $\frac{1}{4^8}$, we're canceling out eight of the 4s in the multiplication. This leaves us with 4 multiplied by itself three times, which is exactly what $4^3$ means. So, our answer makes perfect sense! We've not only simplified the expression using the product of powers rule, but we've also confirmed our answer by understanding the underlying concepts of exponents and reciprocals.

Calculating the Final Value

Alright, we've successfully rewritten the expression $(4^{11})(4^{-8})$ in the form $4^n$, and we've found that n is 3. So, we have $4^3$. But let's take it a step further and actually calculate the final value. This will give us a concrete number that we can understand and visualize. Calculating the final value of an exponential expression is straightforward. It simply means performing the repeated multiplication that the expression represents. In this case, $4^3$ means 4 multiplied by itself three times: 4 * 4 * 4. Let's break it down step by step. First, we multiply the first two 4s: 4 * 4 = 16. Now, we take that result (16) and multiply it by the remaining 4: 16 * 4 = 64. And there we have it! The final value of $4^3$ is 64. So, we can confidently say that $(4^{11})(4^{-8}) = 64$. This simple calculation brings the abstract concept of exponents down to earth. We started with a somewhat complex expression involving exponents, applied the product of powers rule to simplify it, and then calculated the final value. This process highlights the power and elegance of mathematical tools. By understanding the rules of exponents and how to apply them, we can transform seemingly difficult problems into manageable calculations. But it's not just about getting the right answer. It's also about understanding why the answer is what it is. That's why we took the time to break down the process step by step, explaining the logic behind each step. This deeper understanding is what truly empowers us to tackle more complex mathematical challenges. So, next time you encounter an exponential expression, remember the product of powers rule, remember how to handle negative exponents, and remember the fundamental concept of repeated multiplication. With these tools in your arsenal, you'll be able to simplify and solve a wide range of problems with confidence. And don't forget to practice! The more you work with exponents, the more comfortable you'll become with them. So, keep exploring, keep experimenting, and keep learning! Math is a journey, not a destination, and every step you take brings you closer to a deeper understanding of the world around us. Now, let's summarize what we've learned today and reinforce the key concepts.

Summary and Key Takeaways

Alright, guys, let's recap what we've covered today. We started with the expression $(4^{11})(4^{-8})$ and successfully rewrote it in the form $4^n$. Along the way, we learned some crucial concepts about exponents and how to manipulate them. The key takeaway here is the product of powers rule, which states that when you multiply two exponential expressions with the same base, you can add their exponents: $a^m * a^n = a^(m+n)$. This rule is a powerful tool for simplifying expressions and making calculations easier. We also revisited the concept of negative exponents. Remember, a negative exponent indicates the reciprocal of the base raised to the positive version of that exponent. For example, $4^{-8}$ is the same as $\frac{1}{4^8}$. Understanding negative exponents is essential for working with fractions and small values in exponential form. Furthermore, we emphasized the importance of understanding the underlying concepts behind the rules. It's not enough to just memorize the rules; you need to understand why they work. This deeper understanding allows you to apply the rules in different contexts and solve a wider range of problems. We also highlighted the importance of step-by-step problem-solving. By breaking down a complex problem into smaller, manageable steps, we can avoid errors and gain a clearer understanding of the process. Each step builds upon the previous one, leading us to the final solution. Finally, we emphasized the value of practice. Like any skill, mathematical proficiency requires consistent practice. The more you work with exponents, the more comfortable and confident you'll become. So, don't be afraid to tackle challenging problems and experiment with different approaches. In conclusion, we've not only solved a specific problem but also gained a deeper understanding of exponential expressions and the rules that govern them. We've learned how to apply the product of powers rule, how to handle negative exponents, and how to break down complex problems into manageable steps. With these skills in your toolkit, you're well-equipped to tackle a wide range of mathematical challenges. So, keep learning, keep practicing, and keep exploring the fascinating world of mathematics! Remember, every problem is an opportunity to learn something new and strengthen your understanding. And with that, we've reached the end of our journey for today. But the world of mathematics is vast and full of exciting discoveries, so keep exploring and keep learning!

$\boxed{4^3}$ or $\boxed{64}$