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

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Hey guys! Ever stumbled upon an algebraic expression that looks like a monster? Don't worry, we've all been there. Today, we're going to break down a seemingly complex problem: 12a²b x (-3ab) ÷ 9ab². We'll go through it step-by-step so you can not only solve this particular problem but also tackle similar ones with confidence. Math can be fun, especially when you understand the rules of the game! So, let's dive in and make algebra a little less intimidating, shall we?

Understanding the Basics: Order of Operations

Before we even think about those a's and b's, let’s quickly recap the order of operations. Remember PEMDAS/BODMAS? It's the golden rule of math that tells us what to do first. This acronym stands for:

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

This is super important because it dictates the sequence in which we perform calculations. Mess this up, and you'll end up with the wrong answer. In our problem, we have multiplication and division, which we'll handle from left to right. Keeping PEMDAS/BODMAS in mind ensures that we tackle this algebraic expression in the correct order, leading us to the accurate solution. Think of it as the roadmap for solving math problems; it helps us navigate through the steps logically and efficiently. This foundation is crucial for mastering more complex mathematical concepts down the road, so let's always remember our PEMDAS/BODMAS!

Step 1: Multiplication – 12a²b x (-3ab)

Alright, first up, we've got multiplication. We need to multiply 12a²b by -3ab. When multiplying algebraic terms, we multiply the coefficients (the numbers) and then multiply the variables, remembering the rules of exponents. Remember, when multiplying variables with exponents, we add the exponents together. This is a crucial rule to keep in mind. Let’s break it down:

  • Multiply the coefficients: 12 x (-3) = -36
  • Multiply the a terms: a² x a = a^(2+1) = a³
  • Multiply the b terms: b x b = b^(1+1) = b²

So, when we combine these results, 12a²b x (-3ab) equals -36a³b². See? We've already made significant progress! By methodically multiplying the coefficients and then the variables, we've simplified the first part of the expression. This step highlights the importance of understanding the rules of exponents in algebra. Mastering these rules not only helps in solving such problems but also builds a solid foundation for more advanced algebraic manipulations. Now, we're ready to move on to the next operation, but don't forget this crucial multiplication step – it’s the bedrock for what comes next.

Step 2: Division – (-36a³b²) ÷ 9ab²

Now comes the division part. We need to divide our result from Step 1, which is -36a³b², by 9ab². Just like with multiplication, we'll divide the coefficients and then divide the variables, this time subtracting the exponents where applicable. It's like undoing the multiplication we just did. Let’s break it down:

  • Divide the coefficients: -36 ÷ 9 = -4
  • Divide the a terms: a³ ÷ a = a^(3-1) = a²
  • Divide the b terms: b² ÷ b² = b^(2-2) = b⁰ = 1 (Any variable raised to the power of 0 is 1)

So, when we put it all together, (-36a³b²) ÷ 9ab² simplifies to -4a². Notice how the b terms canceled out completely because they had the same exponent. This is a neat trick to remember! This division step showcases the inverse relationship between multiplication and division in algebra. By carefully dividing the coefficients and subtracting the exponents of the variables, we've arrived at the simplified form of the expression. Understanding how to handle division of algebraic terms is crucial for simplifying complex expressions and solving equations. With this step completed, we're one step closer to our final answer.

Final Answer and Conclusion

And there you have it! After performing the multiplication and then the division, we've arrived at the simplified answer: -4a². Wasn't so scary after all, right? By breaking down the problem into smaller, manageable steps and remembering the rules of operations and exponents, we were able to conquer this algebraic challenge.

This example perfectly illustrates the importance of following the order of operations and understanding the fundamental rules of algebra. These skills are not just for solving textbooks problems; they are essential for higher-level math and even many real-world applications. The journey through this problem underscores the power of systematic problem-solving in mathematics. By methodically applying the rules of algebra and arithmetic, we can untangle even the most complex-looking expressions. This approach fosters not only mathematical proficiency but also critical thinking and problem-solving skills that extend far beyond the classroom.

So, the next time you see a similar problem, don’t panic! Just remember our step-by-step guide, and you’ll be able to solve it with confidence. Keep practicing, and you'll become an algebra whiz in no time! Remember, algebra, like any skill, improves with practice. The more you engage with these types of problems, the more comfortable and confident you'll become. So, keep exploring, keep questioning, and keep solving! You've got this!