Solving Complex Mathematical Expressions

Hey guys! Let's dive into solving some tricky mathematical expressions. We've got two big ones here, and we're going to break them down step by step. Think of it like this: we're on a quest to find the hidden numbers at the end of these equations. Grab your thinking caps, and let's get started!

Expression 1: 64 - 3 ⋅ (45 : 5) + (94 - 22) : 9 ⋅ 6

Okay, this first expression looks like a bit of a beast, right? But don't worry, we're going to tame it using the order of operations – you might know it as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Remember, it's all about doing things in the right order.

First up, we tackle the parentheses. Inside the first set of parentheses, we have 45 : 5, which equals 9. Then, in the second set, 94 - 22 gives us 72. So, our expression now looks like this: 64 - 3 ⋅ 9 + 72 : 9 ⋅ 6. See? We're already making progress!

Next, we handle multiplication and division, working from left to right. We've got 3 ⋅ 9, which is 27. Then, 72 : 9 equals 8. And finally, 8 ⋅ 6 gives us 48. Our expression is slimming down: 64 - 27 + 48. We're almost there, guys!

Last but not least, we deal with addition and subtraction, again from left to right. 64 - 27 equals 37. And then, 37 + 48 gives us our final answer: 85. Boom! We solved the first expression. Wasn't so scary after all, right?

So, let's recap the steps for this first expression to make sure we've nailed it:

1. Solve inside the parentheses: (45 : 5) = 9 and (94 - 22) = 72
2. Rewrite the expression: 64 - 3 ⋅ 9 + 72 : 9 ⋅ 6
3. Perform multiplication: 3 ⋅ 9 = 27
4. Perform division: 72 : 9 = 8
5. Perform multiplication: 8 ⋅ 6 = 48
6. Rewrite the expression: 64 - 27 + 48
7. Perform subtraction: 64 - 27 = 37
8. Perform addition: 37 + 48 = 85

The final answer for the first expression is 85. See how breaking it down into smaller steps makes it much more manageable? This is the key to tackling any complex math problem, guys.

Expression 2: 128 ⋅ 146 : 8 ⋅ 9 + 29370 - 19987 + 30540

Alright, now let’s tackle the second expression. It looks even longer, but we'll use the same PEMDAS rules to conquer it. Let's jump right in!

First, we check for parentheses – nope, none here. How about exponents? Nope, not those either. So, we move on to multiplication and division, working from left to right. This is where things get interesting.

We start with 128 ⋅ 146. If you multiply those together, you get 18688. Next up, we have 18688 : 8, which gives us 2336. Then, we multiply 2336 by 9, resulting in 21024. So far, so good! Our expression now looks like this: 21024 + 29370 - 19987 + 30540. We've knocked out the multiplication and division steps like pros!

Now, we're left with addition and subtraction. We work our way from left to right, just like before. First, we add 21024 and 29370, which equals 50394. Then, we subtract 19987 from 50394, giving us 30407. Finally, we add 30540 to 30407, which brings us to our grand total: 60947. Ta-da! We've cracked the code on the second expression!

Let’s quickly recap the steps for this expression:

1. Perform multiplication: 128 ⋅ 146 = 18688
2. Perform division: 18688 : 8 = 2336
3. Perform multiplication: 2336 ⋅ 9 = 21024
4. Rewrite the expression: 21024 + 29370 - 19987 + 30540
5. Perform addition: 21024 + 29370 = 50394
6. Perform subtraction: 50394 - 19987 = 30407
7. Perform addition: 30407 + 30540 = 60947

The final answer for the second expression is a whopping 60947. See, even the long ones are no match for us when we break them down step by step.

Key Takeaways: Mastering the Order of Operations

So, what did we learn today, guys? The biggest takeaway is the importance of the order of operations. PEMDAS is your best friend when it comes to solving complex expressions. Remember:

• Parentheses first
• Exponents next
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

By following these rules, you can tackle any mathematical expression, no matter how intimidating it looks. It’s like having a secret weapon in your math arsenal!

Another key thing we did was break down the problems into smaller, more manageable steps. This is a crucial strategy for problem-solving in general, not just in math. When you're faced with a big, complex task, break it down into smaller parts, and tackle each part one at a time. Before you know it, you'll have conquered the whole thing!

And finally, remember practice makes perfect! The more you work with these types of expressions, the more comfortable and confident you'll become. So, keep practicing, keep breaking things down, and keep that PEMDAS rule in mind. You've got this!

Let's Practice More!

Now that we've conquered these two expressions together, why not try some more on your own? Here are a few practice problems to get you started:

1. 100 - 2 ⋅ (36 : 4) + (120 - 40) : 8 ⋅ 3
2. 256 ⋅ 73 : 4 ⋅ 5 + 15000 - 8500 + 20000
3. 48 + 12 x (25 – 10) ÷ 3 – 16

Remember to follow the PEMDAS rules and break each problem down into manageable steps. Share your answers and your process in the comments below – let's learn together!

Final Thoughts

Math can be challenging, but it can also be super rewarding. Solving complex expressions like these is like cracking a code – it’s a puzzle that needs to be solved. And with the right tools and strategies, you can become a master code-cracker!

So, keep practicing, keep learning, and keep challenging yourselves. You've got the skills, the knowledge, and the determination to conquer any math problem that comes your way. Keep up the great work, guys! You're all doing awesome!