Simplifying Expressions: A Math Breakdown

Oct 22, 2025 by ADMIN 42 views

Hey math enthusiasts! Let's dive into a problem that might seem a bit intimidating at first glance, but trust me, we'll break it down step by step and make it super easy to understand. We're going to tackle the expression $\frac{-18 a^{-2} b^5}{-12 a^{-4} b^{-6}}$ . Our goal is to simplify it and figure out which of the provided options is equivalent to this expression. So, buckle up, grab your pens and paper, and let's get started!

Understanding the Problem: The Basics of Expression Simplification

Alright, guys, before we jump into the nitty-gritty of the problem, let's quickly review the core concepts. Simplifying expressions means we're trying to rewrite a given expression in a more concise or manageable form. This often involves combining like terms, applying exponent rules, and performing basic arithmetic operations. In our case, we're dealing with an expression that involves variables ( $a$ and $b$ ) raised to different powers and coefficients (the numbers in front of the variables). The key here is to remember the rules of exponents and how they interact with each other. This is crucial for math expression simplification. We need to focus on what happens when you divide terms with exponents. Remember that when you divide exponents with the same base, you subtract their powers. For example, $x^m / x^n = x^{(m-n)}$ .

Our problem has negative signs and coefficients, so we'll have to deal with them first. We need to focus on how we treat the coefficients. When you divide coefficients, you just divide the numbers. And then we have the variables ( $a$ and $b$ ), which are raised to various powers. With each variable, we will need to use the rules of exponents to simplify their powers. Don't worry, we'll walk through it slowly. The idea is to transform the original expression, step by step, into a simpler equivalent form, making it easy to see which of the multiple-choice options matches our simplified result. Remember the goal of this task: We must be sure we have simplified the expression as much as possible to match one of the given choices. This kind of problem often appears in algebra and precalculus, and mastering these skills is fundamental for success in higher-level math courses. Always keep in mind the order of operations (PEMDAS/BODMAS) to ensure you perform calculations in the correct sequence. It is also really important to understand the concept of like terms. This means terms that have the same variables raised to the same powers. We can only combine the like terms together. By carefully following these steps, you will be able to simplify and solve this kind of expression confidently!

Step-by-Step Solution: Unraveling the Expression

Now, let's get to the fun part: solving the problem! We'll go through the steps methodically, making sure to explain each one in detail. Our primary expression is $\frac{-18 a^{-2} b^5}{-12 a^{-4} b^{-6}}$ .

Simplify the Coefficients: First, focus on the numbers. We have -18 divided by -12. A negative divided by a negative results in a positive. Thus, $\frac{-18}{-12} = \frac{18}{12}$ . Now, let's simplify this fraction. Both 18 and 12 are divisible by 6. So, $\frac{18}{12} = \frac{3}{2}$ .
Simplify the a terms: Next, let's deal with the a terms. We have $\frac{a^{-2}}{a^{-4}}$ . Remember that when dividing exponents with the same base, we subtract the powers: $a^{-2 - (-4)} = a^{-2 + 4} = a^2$ .
Simplify the b terms: Now, let's look at the b terms. We have $\frac{b^5}{b^{-6}}$ . Again, subtract the powers: $b^{5 - (-6)} = b^{5 + 6} = b^{11}$ .
Combine the Simplified Terms: Finally, let's put it all together. We have the simplified coefficient $\frac{3}{2}$ , the simplified a term $a^2$ , and the simplified b term $b^{11}$ . Combining these, we get $\frac{3}{2} a^2 b^{11}$ .

Therefore, the simplified form of the original expression is $\frac{3 a^2 b^{11}}{2}$ . This is the equivalent expression we were looking for, and it matches one of the multiple-choice options. You did it!

Matching the Solution to the Options: Finding the Equivalent Expression

So, we simplified the expression and got $\frac{3 a^2 b^{11}}{2}$ . Now, let's see which of the given options matches our result. You can see how important simplifying expressions is; it helps us find the right answer. We will examine the given options and compare them with our simplified solution.

Option A: $\frac{2 a^2 b^{11}}{3}$ . This doesn't match because the coefficient is $\frac{2}{3}$ , whereas our answer is $\frac{3}{2}$ .
Option B: $\frac{2 a^2 b^{30}}{3}$ . This doesn't match either, because the powers of b don't align with our result.
Option C: $\frac{3 a^2 b^{11}}{2}$ . This is precisely the same as our simplified expression! The coefficient is $\frac{3}{2}$ , and the powers of a and b are also the same.
Option D: $\frac{3 a^2 b^{30}}{2}$ . Again, the powers of b don't match our solution.

Therefore, the correct answer is Option C. Congratulations! You have successfully simplified the expression and found the correct equivalent one.

Key Takeaways: Mastering Expression Simplification

So, what are the most important things we learned from this exercise? Here's a quick recap to solidify your understanding.

Exponent Rules: Remember the fundamental rules of exponents. When dividing exponents with the same base, subtract the powers. Also, be familiar with how to handle negative exponents.
Coefficient Simplification: Always simplify the coefficients (the numbers) by dividing them and reducing the fraction to its simplest form.
Combining Like Terms: Make sure you're combining like terms correctly. This means ensuring that the variables have the same powers before combining their coefficients.
Order of Operations: Don't forget the order of operations (PEMDAS/BODMAS) to ensure you perform calculations in the right sequence.
Practice, Practice, Practice: The more you practice, the more comfortable you'll become with simplifying expressions. Work through a variety of problems to get the hang of it. You can review similar examples and practice problems in your textbook or online resources.

By following these steps and keeping these key points in mind, you'll be well on your way to mastering expression simplification! Keep practicing, and you'll find that these problems become easier and more enjoyable. Understanding these basics is critical for more advanced mathematical concepts. You will see how these skills are necessary for algebra, trigonometry, calculus, and many more areas of math. So, keep up the great work, and you will continue to see your skills improve. If you have any questions or need further clarification, don't hesitate to ask! Happy simplifying, everyone!