Simplifying Expressions: Finding The Exponent For 'b'

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Hey math enthusiasts! Today, we're diving into a fun algebra problem where we'll help Marina simplify an expression and figure out the correct exponent for the variable 'b'. It's all about understanding how exponents work when dividing terms with variables. Let's break it down, step by step, so you can totally nail similar problems! This is going to be so much fun, guys!

Understanding the Problem: Marina's Simplification

So, the question gives us a fraction: (-4a^-2b^4) / (8a^-6b^-3). Marina has already started simplifying it, and we know that her simplified expression looks like this: -1/2 a^4 b^?. Our mission? To find the missing exponent that should be in place of the question mark for the variable b. This kind of problem is pretty common, so understanding how to tackle it is super important. We're going to use the rules of exponents to get to the answer, and I promise, it's not as scary as it looks. First, we need to understand the initial expression. We are given the following expression: (-4a^-2b^4) / (8a^-6b^-3). Our main goal is to simplify this expression to match Marina's simplified version, and find the exponent for the b variable. We'll meticulously go through each step to make sure we don't miss anything. By the time we're done, you'll be able to confidently solve these problems on your own, no sweat!

Let's get started. We'll start with the coefficients and then move to the variables step by step. This way, we'll keep everything organized and easy to follow. Remember, when you're working with exponents, the key is to be methodical and keep track of each variable. We need to remember that the order of operations is super important! So, let's start with the coefficients, then deal with the variables one by one. This organized approach will keep everything clear, and help us avoid any silly mistakes. Okay, let's keep going. We're well on our way to solving this problem, and soon you'll be a pro at simplifying exponential expressions. Hang in there, and we'll break it down together. I'm going to guide you through this, step by step, so you'll be well-prepared for any exponent problem that comes your way. Get ready to flex those math muscles!

Step-by-Step Simplification: Finding the Exponent

Alright, let's get into the nitty-gritty of simplifying this expression. We'll break it down into smaller, manageable steps to make sure we understand everything perfectly. This is the fun part, where we actually apply the rules of exponents and simplify the expression. We'll go through each term methodically. I am going to explain exactly how to find the missing exponent for b in Marina's simplified expression. This is where the magic happens, so pay close attention. It’s all about applying the rules correctly, and with a bit of practice, you’ll become a master of simplification. Are you ready?

Step 1: Simplify the Coefficients

First things first, let's simplify the coefficients (the numbers). We have -4 in the numerator and 8 in the denominator. When we divide -4 by 8, we get -1/2. This matches the coefficient in Marina's simplified expression, so we're on the right track!

So, we start with our original expression: (-4a^-2b^4) / (8a^-6b^-3). Let's simplify the fraction part first. So, we'll divide the numbers -4 and 8. The negative sign stays, and the fraction simplifies to -1/2. We got this!

Step 2: Simplify the 'a' Terms

Next up, the variable a. We have a^-2 in the numerator and a^-6 in the denominator. When dividing exponents with the same base, you subtract the exponent in the denominator from the exponent in the numerator. So, we get a^(-2 - (-6)) = a^4. This also matches Marina's simplified expression. Keep going guys, we are doing great!

We started with the fraction: (-4a^-2b^4) / (8a^-6b^-3). The expression for a can be simplified as a^-2 / a^-6. Remember that the rule for dividing exponential terms with the same base is to subtract the exponents. So this becomes a^(-2 - (-6)), which simplifies to a^4. So our expression becomes -1/2 * a^4.

Step 3: Simplify the 'b' Terms: The Key to the Answer!

Finally, let's tackle the variable b. We have b^4 in the numerator and b^-3 in the denominator. Again, we subtract the exponents: 4 - (-3) = 7. This means the simplified expression should have b^7. Therefore, the correct exponent for 'b' is 7! The rules are: when dividing exponential terms with the same base, subtract the exponents.

We still have the fraction: (-4a^-2b^4) / (8a^-6b^-3). Now, let's simplify the b terms: b^4 / b^-3. Using the rule, this gives us b^(4 - (-3)). Which simplifies to b^(4+3), or b^7. So the final expression will be -1/2 * a^4 * b^7.

The Answer and Explanation

So, Marina's simplified expression should be -1/2 a^4 b^7. Therefore, the exponent you should use for b is 7. This means the correct answer to the question is NOT A. We got it, guys! The key takeaway here is to remember the rules of exponents: when dividing, subtract the exponents; and when multiplying, add the exponents. Keep practicing, and you'll become a pro in no time! Remember to always break the problem down into small parts, and you'll find the answer much more easily. Keep in mind the rules for exponents, and you'll be well on your way to mastering these kinds of problems. With enough practice, you'll be solving these problems in your sleep!

Why the Other Options are Incorrect

Let's quickly go over why the other options are not correct. Option A suggested an exponent of -7 for b. If we look back at our steps, we can easily see that, following the rules of exponents, we needed to subtract the exponents, which resulted in 4 - (-3) = 7. This shows us the importance of carefully applying the rules and double-checking your work. Always make sure to go step by step, which helps prevent silly mistakes and ensures you get the right answer.

Tips for Solving Similar Problems

Here are some handy tips to help you conquer similar problems in the future.

  • Always start by simplifying the coefficients. This is often the easiest step and sets the foundation for simplifying the variables.
  • Remember the exponent rules! When dividing, subtract the exponents. When multiplying, add the exponents. If you can memorize these simple rules, you will be able to solve most exponent problems.
  • Break the problem down into smaller parts. This makes the problem less intimidating and easier to manage. Just simplify one variable at a time.
  • Double-check your work, especially when dealing with negative exponents. This can save you from making a simple mistake.
  • Practice, practice, practice! The more problems you solve, the more comfortable you'll become with the rules and the easier it will be to solve them. Keep practicing, and you'll be acing these questions in no time!

Conclusion: You've Got This!

Congratulations, guys! You've successfully simplified the expression and found the correct exponent for b. You're well on your way to mastering algebraic expressions and exponents. Keep practicing, and you'll be solving these problems like a pro in no time! Keep up the awesome work!