Solving Algebraic Expressions 12a²b X (-3ab) Divided By 9ab² Step-by-Step

July 30, 2025 by Scholario Team 74 views

Unraveling the Mystery: Solving 12a²b x (-3ab) ÷ 9ab²

Hey guys! Ever stumbled upon a math problem that looks like it's written in another language? Well, today we're going to decode one of those algebraic expressions together. Let's dive into this problem: 12a²b x (-3ab) ÷ 9ab². It might seem daunting at first, but trust me, we'll break it down step by step until it's crystal clear. We'll focus on making sure that by the end of this, you won't just know the answer, but you'll understand why it's the answer. So, buckle up and let's get started!

The Importance of Understanding Algebraic Expressions

Before we jump into solving our specific problem, let's take a moment to appreciate the bigger picture. Algebraic expressions are the backbone of much of mathematics and science. They are the language we use to describe relationships between quantities, model real-world scenarios, and solve complex problems. Understanding how to manipulate and simplify these expressions is not just a classroom exercise; it's a fundamental skill that opens doors to a deeper understanding of the world around us. From physics and engineering to economics and computer science, algebraic expressions are everywhere. So, by mastering these concepts, you're not just acing a test – you're equipping yourself with a powerful tool for future success.

Why This Specific Problem Matters

You might be thinking, "Okay, algebraic expressions are important, but why this particular problem?" Well, this problem, 12a²b x (-3ab) ÷ 9ab², is a fantastic example of how we combine different algebraic operations. It involves multiplication, division, and exponents – all key concepts in algebra. By tackling this problem, we'll reinforce our understanding of these individual operations and, more importantly, how they interact with each other. We'll see how the order of operations matters, how to handle negative signs, and how to simplify expressions with exponents. Plus, the techniques we learn here will be directly applicable to a wide range of similar problems. So, think of this as a stepping stone to conquering more complex algebraic challenges.

Breaking Down the Problem into Manageable Steps

The key to solving any complex problem, especially in math, is to break it down into smaller, more manageable steps. Instead of trying to tackle the entire expression 12a²b x (-3ab) ÷ 9ab² at once, we'll focus on one operation at a time. This not only makes the process less intimidating but also reduces the chances of making mistakes. We'll start with the multiplication, then move on to the division, and finally simplify the result. By taking this step-by-step approach, we'll ensure that we understand each stage of the process and arrive at the correct solution with confidence. Remember, math isn't about rushing to the answer; it's about understanding the journey.

Step-by-Step Solution: Cracking the Code

Alright, let's get down to business and solve this problem! We'll take it slow and steady, explaining each step as we go. Remember, the goal is not just to get the right answer but to understand how we got there. So, let's roll up our sleeves and dive into the step-by-step solution of 12a²b x (-3ab) ÷ 9ab².

Step 1: The Multiplication Magic

The first operation we'll tackle is the multiplication: 12a²b x (-3ab). When multiplying algebraic terms, we need to multiply the coefficients (the numbers in front of the variables) and then multiply the variables. Let's break it down:

Multiply the coefficients: 12 x (-3) = -36. Remember, a positive number multiplied by a negative number results in a negative number.
Multiply the variables: a²b x ab. When multiplying variables with exponents, we add the exponents of the same variable. So, a² x a = a^(2+1) = a³ and b x b = b^(1+1) = b².

Therefore, 12a²b x (-3ab) = -36a³b². We've successfully completed the first step! Now we have a new, slightly simpler expression to work with.

Step 2: Division Time: Dividing and Conquering

Now that we've taken care of the multiplication, we move on to the division: -36a³b² ÷ 9ab². Just like with multiplication, we'll divide the coefficients and then divide the variables. Let's break it down:

Divide the coefficients: -36 ÷ 9 = -4. A negative number divided by a positive number results in a negative number.
Divide the variables: a³b² ÷ ab². When dividing variables with exponents, we subtract the exponents of the same variable. So, a³ ÷ a = a^(3-1) = a² and b² ÷ b² = b^(2-2) = b⁰ = 1 (any variable raised to the power of 0 is equal to 1).

Therefore, -36a³b² ÷ 9ab² = -4a². We're almost there! We've performed the multiplication and division, and now we just need to simplify our result.

Step 3: Simplification Station: Polishing the Answer

In this case, our expression -4a² is already in its simplest form. There are no more like terms to combine or operations to perform. We've done it! We've successfully solved the problem 12a²b x (-3ab) ÷ 9ab² and arrived at the solution: -4a².

Key Concepts Revisited: Reinforcing Your Understanding

Before we celebrate our victory, let's take a moment to recap the key concepts we used to solve this problem. This will help solidify your understanding and make you even more confident in tackling similar problems in the future. We've covered quite a bit, so let's make sure it all sticks!

Order of Operations: The Math Traffic Laws

One of the most crucial things to remember in algebra (and in math in general) is the order of operations. This is like the traffic laws of mathematics – if you don't follow them, you're likely to crash and burn! The order of operations, often remembered by the acronym PEMDAS (or BODMAS), tells us the sequence in which we should perform operations:

Parentheses (or Brackets)
Exponents (or Orders)
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)

In our problem, 12a²b x (-3ab) ÷ 9ab², we first performed the multiplication and then the division because they appear in that order from left to right. If we had performed the division first, we would have arrived at a different (and incorrect) answer. So, always remember PEMDAS (or BODMAS)!

Exponent Rules: Taming the Powers

Exponents are a fundamental part of algebra, and understanding how to work with them is essential. We used two key exponent rules in solving our problem:

Product of Powers: When multiplying variables with exponents, we add the exponents (e.g., a² x a = a^(2+1) = a³).
Quotient of Powers: When dividing variables with exponents, we subtract the exponents (e.g., a³ ÷ a = a^(3-1) = a²).

These rules are essential for simplifying algebraic expressions. Mastering them will make your algebraic journey much smoother and more efficient. Think of them as shortcuts that help you navigate the world of exponents with ease.

Coefficient Handling: Numbers Matter Too

While variables and exponents often get the spotlight, it's important not to forget about the coefficients – the numbers in front of the variables. We need to treat them just like regular numbers, performing the appropriate operations (multiplication, division, etc.) on them. In our problem, we had to multiply and divide the coefficients, paying attention to the signs (positive or negative). Remember, a negative number multiplied or divided by a positive number results in a negative number. These seemingly small details can make a big difference in the final answer.

Practice Makes Perfect: Sharpening Your Skills

Congratulations, guys! You've successfully navigated a complex algebraic problem. But remember, like any skill, algebra requires practice. The more you practice, the more comfortable and confident you'll become. So, don't stop here! Let's talk about how you can keep sharpening your skills.

Online Resources and Worksheets: Your Algebra Toolkit

There are tons of fantastic resources available online that can help you practice algebra. Websites like Khan Academy, Mathway, and Purplemath offer lessons, practice problems, and even step-by-step solutions. You can also find worksheets online that cover a wide range of algebraic topics. These resources can be invaluable tools in your algebra toolkit. They provide a structured way to learn and practice, and they can help you identify areas where you might need extra help.

Seeking Help When Needed: No Shame in Asking

Finally, remember that there's no shame in asking for help when you need it. If you're struggling with a particular concept or problem, reach out to your teacher, classmates, or a tutor. Explaining your difficulties can often help you clarify your understanding, and getting a different perspective can be incredibly helpful. Math is often a collaborative endeavor, so don't hesitate to seek support when you need it.

Conclusion: You've Got This!

Wow, we've covered a lot today! We successfully solved the problem 12a²b x (-3ab) ÷ 9ab², and along the way, we reinforced our understanding of key algebraic concepts like the order of operations, exponent rules, and coefficient handling. We also discussed the importance of breaking down complex problems into smaller steps and the value of practice. Remember, algebra is a journey, not a destination. Keep practicing, keep asking questions, and keep challenging yourself. You've got this!

So, the next time you encounter an algebraic expression that looks intimidating, remember the steps we've discussed. Break it down, take it one operation at a time, and don't be afraid to ask for help. With practice and perseverance, you'll become an algebra master in no time! Keep up the great work, guys!