Simplifying Exponential Expressions: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of exponents and learning how to simplify and evaluate expressions. We're going to break down the expression { \frac{6^{n+5}}{6^{12}} \cdot rac{36^{n+3}}{86} } step-by-step. Don't worry, it might look a little intimidating at first, but trust me, with a little practice, you'll be acing these problems in no time. Let's get started, shall we? This simplifying exponential expressions tutorial will help you understand the core concepts. The key to mastering this is understanding the rules of exponents and applying them methodically. We'll start with the basics and gradually work our way through the problem, explaining each step in detail. So grab your pens and paper, and let's get ready to simplify some expressions! This is going to be fun, I promise! We are going to simplify expressions and make it into something that is easier to work with. These concepts are fundamental in algebra and are used extensively in various fields like physics, engineering, and computer science. By understanding these concepts, you'll be well-equipped to tackle more complex problems later on. We'll explore how to handle fractions with exponents, simplify terms with the same base, and use the power of a power rule. By the end of this guide, you'll have a solid grasp of how to approach and solve these types of problems confidently. The initial expression might seem complex, but breaking it down into smaller, manageable parts makes the simplification process much easier. Think of it like a puzzle; each step brings you closer to the solution. So, let's unlock the secrets of this expression together!

Step 1: Understanding the Problem and Key Concepts

First things first, let's break down the problem. We're dealing with an expression that involves exponents, fractions, and different bases. The main goal here is to simplify exponential expressions to the simplest form possible. Before we jump into the simplification, let's quickly recap some key concepts and rules of exponents that we'll be using throughout this process. You know, just to make sure we're all on the same page, ya know? When we have the same base, when dividing exponents, we subtract the powers. Also, when we have the same base, when multiplying exponents, we add the powers. These are some of the rules we will use. We'll use the quotient rule of exponents (when dividing, subtract the powers), the product rule of exponents (when multiplying, add the powers), and the power of a power rule (when raising a power to another power, multiply the powers). These rules are super important. Understanding these rules is crucial for simplifying exponential expressions effectively. We're also going to need to remember how to handle fractions. Basically, these rules are our tools, and the more familiar you are with them, the easier it will be to solve the problem. Also, remember that a fraction is just a division problem written differently. Understanding these fundamental principles is key to successfully simplifying the given expression. So, keep these rules in mind as we start simplifying the expression. It's like having the secret decoder ring to solve the puzzle!

Rules of Exponents to Remember:

• Quotient Rule: { rac{a^m}{a^n} = a^{m-n} }
• Product Rule: ${ a^m imes a^n = a^{m+n} }$
• Power of a Power Rule: ${ (a^m)^n = a^{m imes n} }$

Step 2: Simplifying the First Fraction

Okay, let's start simplifying the expression step by step. We'll start with the first fraction: { rac{6^{n+5}}{6^{12}} }. Our goal is to simplify expressions by applying the quotient rule of exponents. According to the quotient rule, when dividing exponential expressions with the same base, we subtract the exponents. In this case, our base is 6. So, we'll subtract the exponent in the denominator (12) from the exponent in the numerator (n+5). So, we have ${ 6^{(n+5) - 12} }$. Let's simplify that exponent: ${ n + 5 - 12 = n - 7 }$. So the first fraction simplifies to ${ 6^{n-7} }$. See? Not so bad, right? We've managed to make the expression a little cleaner by combining those exponents. By applying the quotient rule, we've significantly reduced the complexity of the first part of the expression. Now that we've simplified this part, we can move on to the next one. We're on our way to solving this, folks! Always remember to keep your work organized and to write down each step clearly. This helps you avoid silly mistakes and makes it easier to spot any errors if you need to go back and check your work. And that's it for the first part. Let's keep going.

Step 3: Simplifying the Second Fraction

Alright, let's turn our attention to the second part of the expression: { rac{36^{n+3}}{86} }. Now, this fraction is a little different because the bases aren't the same. But we can change that! We know that 36 is ${ 6^2 }$. So, we can rewrite the expression as { rac{(6^2)^{n+3}}{86} }. Now we can apply the power of a power rule. The power of a power rule states that when you have an exponent raised to another exponent, you multiply the exponents. So, we'll multiply 2 by (n+3). This gives us ${ 2 imes (n+3) = 2n + 6 }$. Therefore, our second fraction simplifies to { rac{6^{2n+6}}{86} }. We're making progress. We're making the expression simpler and easier to work with. Remember that our goal is to evaluate expressions and get a more manageable form. Always keep your eye on the prize - the simpler, the better. We've got this, guys! This step shows how you can change the bases so they match and you can simplify them. We are using the correct rules of exponents. We are just about there to solving this problem. Keep up the good work and we'll have it done in no time.

Step 4: Combining the Simplified Fractions

Okay, now we have simplified both fractions. The expression now looks like this: { 6^{n-7} imes rac{6^{2n+6}}{86} }. We're getting closer to solving this problem! The goal now is to put all our findings together, simplifying the entire expression. Notice that we now have multiplication. We have two exponential terms with the same base (6). So, let's apply the product rule of exponents. According to the product rule, when multiplying exponential expressions with the same base, you add the exponents. Therefore, we will be adding ${n - 7}$ and ${2n + 6}$. So, ${(n - 7) + (2n + 6) = 3n - 1}$. Therefore, we combine the exponents to get: ${ 6^{3n-1} }$. Our expression is now { rac{6^{3n-1}}{86} }. Now, it does look pretty simple, yeah? We're taking the expression and making it smaller. Also, notice that the denominator has no exponent or variable in it. That cannot be simplified any further, which means we are done simplifying! Our final expression is now { rac{6^{3n-1}}{86} }. This is the simplified form of our original expression. That is the answer, guys! Great job. From a complex-looking expression to a much cleaner form! We've made it! This last part involves adding the exponents after we combine the fractions. Always double-check your work to make sure you have applied all rules of exponents correctly and that there are no mistakes. We've simplified the expression. It is a simplified exponential expression. That's all there is to it.

Step 5: Final Answer and Conclusion

So, after all the hard work, the simplified form of the expression { rac{6^{n+5}}{6^{12}} imes rac{36^{n+3}}{86} } is { rac{6^{3n-1}}{86} }. Awesome job, everyone! We've successfully simplified the expression step-by-step. Remember, the key is to break down the problem, apply the rules of exponents correctly, and take it one step at a time. The rules are your friends. If you still have the same base, apply the rules of exponents. Also, always keep the original problem in mind. We want to simplify the expression to its simplest form. We've covered the quotient rule, the product rule, and the power of a power rule. We've also learned how to rewrite bases to make them match. Keep practicing these types of problems, and you'll become more confident in your ability to simplify exponential expressions. Great job everyone! You're now one step closer to mastering exponents. Keep practicing, and you'll be a pro in no time! Remember that practice makes perfect, so keep practicing these problems to sharpen your skills. With consistent effort, you'll become adept at simplifying and evaluating even the most complex exponential expressions! Congratulations on successfully simplifying the expression! Keep up the excellent work, and I'll see you next time!