Simplifying Algebraic Expressions Equivalent Form Of (-18 A-2 B5) / (-12 A-4 B-6)

Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of algebraic expressions, specifically focusing on simplifying fractions with exponents. We'll be tackling a problem that might seem daunting at first glance, but with a step-by-step approach and a dash of algebraic wizardry, we'll conquer it together. Our mission is to find the expression equivalent to $\frac{-18 a^{-2} b^5}{-12 a^{-4} b^{-6}}$, assuming that $a$ and $b$ are not equal to zero. So, buckle up, grab your calculators (just kidding, we won't need them!), and let's embark on this mathematical journey!

Understanding the Problem

Before we jump into the solution, let's break down the problem and make sure we understand what we're dealing with. We have a fraction where both the numerator and the denominator contain variables ($a$ and $b$) raised to different powers, some of which are negative. The key here is to remember the rules of exponents, especially how to handle negative exponents and how to simplify fractions. Remember, a negative exponent means we're dealing with the reciprocal of the base raised to the positive exponent. For example, $x^{-n} = \frac{1}{x^n}$. This little nugget of knowledge is going to be super helpful in simplifying our expression.

Our goal is to manipulate the given expression, $\frac{-18 a^{-2} b^5}{-12 a^{-4} b^{-6}}$, using the rules of exponents and fraction simplification until we arrive at one of the answer choices provided. The answer choices are:

A. $\frac{2 a^2 b^{11}}{3}$ B. $\frac{2 a^2 b^{30}}{3}$ C. $\frac{3 a^2 b^{11}}{2}$ D. $\frac{3 a^2 b^{30}}{2}$

By carefully applying the rules of exponents and simplifying the coefficients, we'll be able to match our simplified expression with the correct answer choice. It's like a mathematical puzzle, and we're about to solve it! So, let's roll up our sleeves and get started.

Step-by-Step Solution

Okay, let's get down to business and simplify the expression. Remember, the key to success in math (and in life, really) is to break down complex problems into smaller, manageable steps. We'll tackle this expression one piece at a time, making sure we understand each step along the way.

1. Simplify the Coefficients

First, let's focus on the numerical coefficients: -18 and -12. We have a fraction $\frac{-18}{-12}$. A negative divided by a negative is a positive, so we can rewrite this as $\frac{18}{12}$. Now, we need to find the greatest common divisor (GCD) of 18 and 12, which is 6. We can divide both the numerator and the denominator by 6 to simplify the fraction: $\frac{18 \div 6}{12 \div 6} = \frac{3}{2}$. So, the numerical part of our simplified expression will be $\frac{3}{2}$. Awesome! We've already made significant progress.

2. Simplify the 'a' terms

Next up are the 'a' terms. We have $a^{-2}$ in the numerator and $a^{-4}$ in the denominator. When dividing terms with the same base, we subtract the exponents. So, we have $a^{-2} / a^{-4} = a^{-2 - (-4)} = a^{-2 + 4} = a^2$. Fantastic! The 'a' terms have simplified to $a^2$.

3. Simplify the 'b' terms

Now, let's tackle the 'b' terms. We have $b^5$ in the numerator and $b^{-6}$ in the denominator. Again, we subtract the exponents: $b^5 / b^{-6} = b^{5 - (-6)} = b^{5 + 6} = b^{11}$. Excellent! The 'b' terms have simplified to $b^{11}$.

4. Combine the Simplified Terms

We've simplified the coefficients, the 'a' terms, and the 'b' terms. Now, let's put it all together. We have $\frac{3}{2}$ from the coefficients, $a^2$ from the 'a' terms, and $b^{11}$ from the 'b' terms. Combining these gives us $\frac{3 a^2 b^{11}}{2}$.

5. Match with Answer Choices

Finally, let's compare our simplified expression, $\frac{3 a^2 b^{11}}{2}$, with the answer choices provided. We can see that it matches answer choice C. Woohoo! We've successfully simplified the expression and found the equivalent form.

Common Mistakes to Avoid

Simplifying algebraic expressions can be tricky, and it's easy to make mistakes if you're not careful. Let's talk about some common pitfalls to avoid so you can ace these types of problems.

Forgetting the Rules of Exponents

The most common mistake is messing up the rules of exponents. Remember, when dividing terms with the same base, you subtract the exponents, not divide them. And when you have a negative exponent, it means you're dealing with the reciprocal. Review these rules regularly, and you'll be golden.

Incorrectly Handling Negative Signs

Negative signs can be sneaky! Make sure you're paying close attention to them, especially when subtracting exponents or simplifying fractions with negative numbers. A small mistake with a negative sign can throw off your entire answer. Double-check your work to ensure you've handled them correctly. Always double check the signs! It's a simple step, but it can save you from making a silly mistake.

Not Simplifying Completely

Sometimes, you might simplify part of the expression but forget to simplify it completely. For example, you might simplify the 'a' terms and the 'b' terms but forget to simplify the numerical coefficients. Always make sure you've simplified every part of the expression as much as possible before comparing it to the answer choices.

Rushing Through the Problem

Math problems require focus and attention to detail. Rushing through the problem can lead to careless errors. Take your time, write out each step clearly, and double-check your work as you go. It's better to be slow and accurate than fast and wrong. Think of it like baking a cake – you can't rush the process if you want it to turn out delicious!

By being aware of these common mistakes and taking steps to avoid them, you'll be well on your way to mastering algebraic expressions. Keep practicing, and you'll become a simplification superstar!

Practice Problems

To truly master simplifying algebraic expressions, practice is key! Here are a few more problems for you to try. Working through these will help solidify your understanding of the concepts and build your confidence. Remember, the more you practice, the easier it will become.

1. Simplify $\frac{24 x^3 y^{-2}}{-16 x^{-1} y^4}$
2. Simplify $\frac{-35 a^{-5} b^7}{14 a^2 b^{-3}}$
3. Simplify $\frac{15 m^4 n^{-1}}{-25 m^{-2} n^5}$

Try solving these problems on your own, using the steps we discussed earlier. Don't be afraid to make mistakes – they're a natural part of the learning process. And if you get stuck, review the steps and explanations we've covered. The answers to these problems are provided below, but try to solve them without looking first!

Real-World Applications

Now, you might be thinking,