Simplifying Exponential Expressions: A Step-by-Step Guide

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Hey math enthusiasts! Today, we're diving into the world of exponents and simplifying expressions. We'll break down the expression: 9−12×8134912\frac{9^{-\frac{1}{2}} \times 81^{\frac{3}{4}}}{9^{\frac{1}{2}}}. Don't worry if it looks a little intimidating at first. We'll go through it step by step, making sure everyone understands the process. This is a great exercise to brush up on your exponent rules and practice simplifying complex expressions. So, grab your calculators (optional!), and let's get started. By the end of this, you'll be a pro at simplifying exponential expressions! I will use detailed explanations and break down each step so that you understand the process of calculation of expressions. Let's make sure we understand the basic rules of exponents first. Remember, a negative exponent means taking the reciprocal, and fractional exponents represent roots. We'll apply these rules to simplify the expression and arrive at our final answer. Ready? Let's go!

Step-by-Step Simplification

Step 1: Handling Negative Exponents

First things first, let's address the negative exponent in our expression. We have 9−129^{-\frac{1}{2}}. Remember that a negative exponent indicates the reciprocal of the base raised to the positive value of the exponent. So, 9−129^{-\frac{1}{2}} is the same as 1912\frac{1}{9^{\frac{1}{2}}}. The expression now looks like this: 1912×8134912\frac{\frac{1}{9^{\frac{1}{2}}} \times 81^{\frac{3}{4}}}{9^{\frac{1}{2}}}. See? Not so scary, right? By understanding what the negative exponent means, you've made a huge step in simplifying this expression. We are essentially rewriting the expression to get rid of the negative exponents. This makes the next steps easier to manage and understand. This process is about translating and applying the basic rules we know. Always keep the exponent rules in mind, and you will be able to solve these types of problems in a breeze. That's the first step to conquering this expression. Let's move on to the next one.

Step 2: Dealing with Fractional Exponents

Next up, let's tackle the fractional exponents. We have 9129^{\frac{1}{2}} and 813481^{\frac{3}{4}}. Remember that a fractional exponent like 12\frac{1}{2} represents a square root. So, 9129^{\frac{1}{2}} is the square root of 9, which is 3. Now, let's look at 813481^{\frac{3}{4}}. This can be interpreted as the fourth root of 81, raised to the power of 3. The fourth root of 81 is 3 (since 3×3×3×3=813 \times 3 \times 3 \times 3 = 81), and then we cube it. 333^3 is 27. So, the expression becomes: 13×273\frac{\frac{1}{3} \times 27}{3}. See, things are getting simpler by the moment! Understanding how fractional exponents work is key to making this problem solvable. Many students stumble on this part, but once you get the hang of it, these types of problems are straightforward. Remember, practice makes perfect, so keep solving these types of problems. You'll become a fractional exponent master in no time! You're doing great, keep it up.

Step 3: Simplifying the Numerator

Now, let's simplify the numerator. We have 13×27\frac{1}{3} \times 27. Multiply 13\frac{1}{3} by 27. That's the same as dividing 27 by 3, which equals 9. So, the expression simplifies to 93\frac{9}{3}. We are slowly but surely getting closer to the final answer. Each step is designed to reduce the complexity of the expression. You're applying basic arithmetic here, so make sure you don't get stuck here. The previous steps have set you up perfectly for this. By simplifying the numerator first, we are making the last step, which is division, easier to solve. Always remember the order of operations, and you'll navigate these problems with ease. This step highlights the importance of combining the various terms to work toward the ultimate answer. Keep your focus on each step, and you will be done with the problem in no time. This is where it all comes together! Almost there!

Step 4: Final Calculation

Finally, we have 93\frac{9}{3}. Divide 9 by 3, and you get 3. Therefore, the simplified expression is equal to 3. Congratulations, you've successfully simplified the expression! That wasn't so bad, was it? We started with a complex-looking expression, but by breaking it down step by step and applying the exponent rules, we arrived at a simple answer. Give yourself a pat on the back. You've earned it! It's rewarding to see how complex problems can be simplified with the right approach. The key is to understand each step. We started with understanding the basics, then progressed to negative and fractional exponents, and, finally, we simplified the expression. It's a great illustration of how understanding the fundamentals of mathematics can allow you to tackle more complex problems. You can use the approach to solve other problems as well. So, embrace the challenge, and remember that with consistent practice, you can master any mathematical concept.

Key Takeaways and Tips

Let's recap what we've learned and provide some useful tips for similar problems. This entire exercise helps us to understand and apply the rules of exponents. Understanding Exponent Rules is the foundation. Make sure you understand how to handle negative exponents, fractional exponents, and the general rules of exponentiation. Practice Makes Perfect. The more you practice, the more comfortable you'll become with simplifying expressions. Work through various examples to reinforce your understanding. Always apply the order of operations (PEMDAS/BODMAS). This will ensure that you simplify the expression in the correct order. The use of calculators is allowed, but try to do the steps by hand. This will improve your understanding. Don't be afraid to break down the problem into smaller steps. This will make it easier to manage and less intimidating. Write out each step clearly. This helps you to avoid errors and track your progress. Don't give up! Simplifying expressions can be challenging, but with patience and practice, you can master it. Keep a reference sheet with the exponent rules handy. This will help you quickly recall the rules when solving problems. Now, go forth and conquer those exponential expressions!

Additional Examples

To solidify your understanding, let's work through a few more examples. These examples are designed to build your confidence and offer further practice. I will give you a few more examples so you can practice more. Always try to understand the process. The process is more important than the solution, so by solving a few examples, you'll be more confident. Remember, practice is the secret to mastering mathematical concepts. Let's get started with more examples! Here are a couple more problems for you to practice, along with their solutions. Try solving them on your own before looking at the solutions. This will test your understanding.

Example 1:

Simplify 4−12×1634212\frac{4^{-\frac{1}{2}} \times 16^{\frac{3}{4}}}{2^{\frac{1}{2}}}

  • Solution:
    1. 4−12=1412=124^{-\frac{1}{2}} = \frac{1}{4^{\frac{1}{2}}} = \frac{1}{2}
    2. 1634=(1614)3=23=816^{\frac{3}{4}} = (16^{\frac{1}{4}})^3 = 2^3 = 8
    3. 12×8212=42=22\frac{\frac{1}{2} \times 8}{2^{\frac{1}{2}}} = \frac{4}{\sqrt{2}} = 2\sqrt{2}

Example 2:

Simplify 25−12×12523512\frac{25^{-\frac{1}{2}} \times 125^{\frac{2}{3}}}{5^{\frac{1}{2}}}

  • Solution:
    1. 25−12=12512=1525^{-\frac{1}{2}} = \frac{1}{25^{\frac{1}{2}}} = \frac{1}{5}
    2. 12523=(12513)2=52=25125^{\frac{2}{3}} = (125^{\frac{1}{3}})^2 = 5^2 = 25
    3. 15×25512=55=5\frac{\frac{1}{5} \times 25}{5^{\frac{1}{2}}} = \frac{5}{\sqrt{5}} = \sqrt{5}

Keep practicing, and soon you'll be simplifying these expressions without a second thought. Remember, mastering these concepts takes time and effort. Each problem is a stepping stone toward becoming proficient in mathematics. Happy calculating, and keep up the great work! You've got this!