Evaluating -9 × (-9 ÷ 3 + 7) A Step-by-Step Guide

by James Vasile 50 views

Hey guys! Today, we're diving into the world of mathematical expressions, and we're going to break down how to evaluate one step by step. This particular expression involves a mix of multiplication, division, addition, and negative numbers, so it's a great example to help us understand the order of operations. We'll be tackling this expression: -9 × (-9 ÷ 3 + 7). Don't worry, it might look intimidating at first, but we'll break it down into manageable chunks and you'll be a pro in no time!

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we jump into the nitty-gritty, let's quickly recap the order of operations. This is the golden rule that dictates the sequence in which we perform mathematical operations. You might have heard of the acronyms PEMDAS or BODMAS, which are just handy ways to remember the order. They stand for:

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

Basically, we need to solve anything inside parentheses first, then exponents, then multiplication and division (working from left to right), and finally addition and subtraction (also from left to right). Ignoring this order can lead to completely wrong answers, so it's super important to get it right. It's like following a recipe – you need to add the ingredients in the correct order to bake a delicious cake!

Think of PEMDAS/BODMAS as your roadmap for solving any mathematical expression. It guides you through the steps, ensuring you arrive at the correct destination – the solution! When we approach our expression, -9 × (-9 ÷ 3 + 7), we'll be using this roadmap every step of the way. Remember, mathematical expressions are like puzzles; each step is a piece that fits into the bigger picture. By following the order of operations, we're essentially fitting the pieces together in the right way.

Now, let’s talk a little more about why this order is so vital. Imagine if we just performed the operations in the expression from left to right, without considering PEMDAS/BODMAS. We might end up doing the multiplication before the division inside the parentheses, which would completely throw off the result. The order of operations ensures consistency and accuracy in mathematics. It's a universal rule that everyone follows, so when we see an expression like -9 × (-9 ÷ 3 + 7), we all know exactly how to approach it.

Step 1: Tackling the Parentheses (-9 ÷ 3 + 7)

Alright, let's get our hands dirty with our expression! According to PEMDAS/BODMAS, our first mission is to conquer the parentheses. Inside the parentheses, we have -9 ÷ 3 + 7. Notice that we have two operations here: division and addition. Which one do we tackle first? You guessed it – division! Remember, multiplication and division take precedence over addition and subtraction.

So, let's focus on -9 ÷ 3. Dividing a negative number by a positive number gives us a negative result. In this case, -9 divided by 3 is -3. Think of it as splitting a debt of 9 dollars equally among 3 people; each person owes 3 dollars. Now our expression inside the parentheses looks like this: -3 + 7.

Moving on, we now have a simple addition problem. We're adding a negative number (-3) to a positive number (7). You can think of this as having 7 dollars and owing 3 dollars. After paying off your debt, you'd have 4 dollars left. So, -3 + 7 = 4. The value inside the parentheses is now simplified to 4. That wasn't so bad, right? We've successfully navigated the first step by prioritizing division within the parentheses and then taking care of the addition. This methodical approach is key to solving more complex expressions as well.

Think of this step like preparing the ingredients before cooking. We're simplifying the components within the parentheses before we move on to the main dish. Each part of the expression needs to be carefully evaluated before we combine it with other parts. By breaking down the parentheses first, we’ve made the overall expression much easier to handle. We've essentially isolated the most complex part and dealt with it, making the rest of the calculation smoother. So, remember, when you see parentheses, that’s your starting point. Solve what’s inside them, following the order of operations within the parentheses themselves.

Step 2: Multiplication -9 × 4

Now that we've simplified the expression inside the parentheses to 4, our original expression looks much cleaner: -9 × 4. We're left with a single operation: multiplication. This is the final hurdle before we reach our answer! We're multiplying a negative number (-9) by a positive number (4). Just like with division, multiplying a negative number by a positive number results in a negative answer.

So, let's multiply the absolute values first: 9 multiplied by 4 is 36. Since we're multiplying a negative by a positive, our final result will be negative. Therefore, -9 × 4 = -36. And there you have it! We've successfully evaluated the expression. The answer is -36.

Think of this final step as the grand finale of our mathematical journey. We've simplified the expression piece by piece, and now we're bringing it all together with a single, decisive multiplication. Each step has led us to this point, and the result, -36, is the culmination of our efforts. It's like the final brushstroke on a painting, completing the masterpiece. This step highlights the importance of understanding the rules of multiplying negative and positive numbers. A simple sign error can throw off the entire calculation, so it's crucial to pay close attention to those details. We’ve navigated the parentheses, handled the division and addition, and now we’ve nailed the multiplication, bringing us to the final answer.

Final Answer: -36

We've successfully evaluated the expression -9 × (-9 ÷ 3 + 7), and the final answer is -36. Awesome job, guys! We took a seemingly complex expression and broke it down into manageable steps, using the order of operations (PEMDAS/BODMAS) as our guide. We first tackled the parentheses, simplifying the division and addition within. Then, we performed the final multiplication to arrive at our result.

Remember, the key to mastering mathematical expressions is to practice consistently and to follow the order of operations diligently. It's like learning a new language – the more you practice, the more fluent you become. So, keep practicing, keep asking questions, and you'll become a math whiz in no time!

I hope this step-by-step explanation was helpful. Remember, mathematics is not just about memorizing formulas, it's about understanding the process and applying the concepts. By breaking down problems into smaller steps, we can tackle even the most challenging expressions. Keep up the great work, guys, and happy problem-solving!