Math Problem: Order Of Operations With Commutative Properties

Hey guys! Let's dive into some math problems that involve using the commutative and associative properties to make calculations easier. These properties are super handy because they let us rearrange and regroup numbers in addition and multiplication without changing the result. This can save us a lot of time and effort, especially when dealing with larger numbers or fractions. We’ll break down each problem step-by-step, so you can see exactly how these properties work in action.

Understanding Commutative and Associative Properties

Before we jump into the problems, let’s quickly recap what the commutative and associative properties are. The commutative property states that the order in which you add or multiply numbers doesn't change the answer. For example, a + b = b + a and a * b = b * a. The associative property says that the way you group numbers in addition or multiplication doesn't change the result. That is (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c). Knowing these rules allows us to reorganize the terms in an expression to make it easier to compute, especially when we have a mix of positive and negative numbers or fractions. By strategically rearranging the order and grouping, we can often find pairs of numbers that are easier to work with, such as those that add up to a round number or simplify fractions. This not only speeds up the calculation process but also reduces the chances of making errors. Now, let’s tackle those problems!

Problem 1: 34

Okay, let's start with the first one: 34. Well, there's not much to do here since it's just a single number! It seems like this might be part of a bigger problem that's missing. But if we're just looking at 34, then 34 is our answer. Sometimes in math, you get a simple one just to ease you in. Think of it as a warm-up before the main workout. We can't really apply commutative or associative properties here because there's nothing to reorder or regroup. It's just a standalone number, patiently waiting for its turn to be part of a more complex equation. So, for now, we acknowledge its presence and move on to the next challenge, where things are sure to get a bit more interesting and we can actually put those properties to good use. Remember, math isn't always about complex calculations; sometimes, it's about recognizing the simple elements within a larger problem.

Problem 2: 353 * (-8) * 3/3 * 6 * 13

Now, this looks more like it! We’ve got a string of multiplications here: 353 * (-8) * 3/3 * 6 * 13. The key here is to use both the commutative and associative properties to make our lives easier. First off, remember that 3/3 is just 1, so we can simplify that right away. Now we have 353 * (-8) * 1 * 6 * 13. Next, let's rearrange the terms to group the easier numbers together. How about we put the 6 and -8 together? This gives us 353 * (-8 * 6) * 13. Now, -8 * 6 is -48, so we have 353 * (-48) * 13. This is where a calculator might come in handy, but we've already made it simpler by regrouping. Multiplying these numbers together, we get -220368. So, by using those properties, we broke down a seemingly complicated problem into manageable chunks. Remember, guys, the goal isn't just to get the right answer, but to get there efficiently!

Problem 3: -595 * (-11) * 11/13 * 3

Alright, let's tackle this one: -595 * (-11) * 11/13 * 3. Again, we're going to use our trusty commutative and associative properties to simplify things. First, notice we have a couple of negative numbers. Remember that a negative times a negative is a positive, so -595 * (-11) is the same as 595 * 11. Let's rewrite the problem as 595 * 11 * 11/13 * 3. Now, let's regroup to make the multiplication easier. How about we pair 11 and 3 together? This gives us 595 * (11 * 3) * 11/13. Now we have 595 * 33 * 11/13. We can see that 595 is divisible by 5 and also by 7*17 and 13 is a prime number. Let's calculate 595 * 33, which equals 19635. Now we have 19635 * 11/13. This simplifies to 215985 / 13, which equals 16614.23 (approximately). See how breaking it down step by step makes it less intimidating? Remember to keep an eye out for opportunities to simplify early on, like we did with the negative signs. This can save you from bigger headaches later!

Problem 4: 11½ - (-5) - (-21³)

Okay, guys, this one looks a bit different because we're dealing with subtraction and an exponent: 11½ - (-5) - (-21³). First things first, let's deal with that mixed number. 11½ is the same as 11.5. Now, let's handle those negative signs. Subtracting a negative is the same as adding, so - (-5) becomes + 5, and - (-21³) becomes + 21³. So, our problem now looks like this: 11.5 + 5 + 21³. Next, we need to calculate 21³, which means 21 * 21 * 21. That's 9261. Now our problem is 11.5 + 5 + 9261. This is a simple addition problem now. 11.5 + 5 is 16.5, and then 16.5 + 9261 is 9277.5. So, the answer is 9277.5. The key here was to take it one step at a time: convert the mixed number, deal with the negatives, calculate the exponent, and then add everything up. Remember, breaking down a problem into smaller steps makes it much less scary!

Problem 5: 3 * (-13) * 223

Let's dive into this multiplication problem: 3 * (-13) * 223. We're going to use the commutative property to rearrange the numbers to make it easier. First, let's multiply the smaller numbers: 3 * (-13). That gives us -39. Now our problem is -39 * 223. This is a straightforward multiplication problem. When we multiply -39 by 223, we get -8697. So, the answer is -8697. By multiplying the smaller numbers first, we made the overall calculation simpler. Always look for those opportunities to make your life easier, guys! It's all about working smarter, not harder, in math.

Problem 6: 8 * (-9) * (1) * 10

Alright, let's wrap things up with this final problem: 8 * (-9) * (1) * 10. We can use the commutative and associative properties here to make this calculation a breeze. Remember, multiplying by 1 doesn’t change the number, so we can focus on the other terms. Let’s rearrange the numbers to group the ones that are easy to multiply together. How about we put the 8 and 10 together? This gives us (8 * 10) * (-9) * 1. Now, 8 * 10 is 80, so we have 80 * (-9) * 1. Next, let's multiply 80 by -9. That's -720. Finally, we have -720 * 1, which is simply -720. So, the answer is -720. By strategically regrouping and multiplying, we turned a potentially messy problem into a quick calculation. And that's how you use those properties to your advantage!

Final Thoughts

So, there you have it, guys! We’ve tackled a bunch of math problems using the commutative and associative properties. Remember, these properties are your friends in math. They allow you to rearrange and regroup numbers to make calculations easier and less prone to errors. Whether you're dealing with multiplication, addition, or a mix of operations, keep an eye out for opportunities to simplify using these properties. Math can be challenging, but with the right tools and techniques, you can conquer any problem. Keep practicing, and you’ll become a math whiz in no time!