Calculate Math Expressions: Step-by-Step Solutions

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Hey guys! Let's break down these math problems together, step by step. We've got four expressions to calculate, and we'll make sure to explain each one clearly so everyone can follow along. Ready to dive in? Let's get started!

a) Calculating 2017 + 2017 * 3

When we tackle math expressions, it's super important to remember our order of operations. You might have heard of PEMDAS/BODMAS, which tells us the sequence we need to follow: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In this first expression, 2017 + 2017 * 3, we need to do the multiplication before we do the addition. This is a crucial step to get the correct answer.

First, let's focus on the multiplication part: 2017 * 3. If you multiply these two numbers, you'll find that 2017 multiplied by 3 equals 6051. So now, our expression looks like this: 2017 + 6051. We've simplified the multiplication part, and we're one step closer to the final answer. Remember, getting the order right is half the battle in these calculations!

Now that we've handled the multiplication, the next step is to add 2017 and 6051. When you add these two numbers together, 2017 plus 6051 comes out to 8068. This is the final result for the first expression. See? It's not so bad when we break it down into smaller, manageable steps. By following the order of operations, we've nailed this calculation. So, the final answer for part (a) is a solid 8068. Keep this in mind as we move on to the next expression!

To summarize, here’s the breakdown:

  1. Multiplication: 2017 * 3 = 6051
  2. Addition: 2017 + 6051 = 8068

Therefore, 2017 + 2017 * 3 = 8068.

b) Calculating 2017 + 2017 * 2017 + 2018

Alright, let's jump into the next expression: 2017 + 2017 * 2017 + 2018. Just like before, we need to stick to the order of operations. This means we'll be doing the multiplication first. We've got 2017 multiplied by itself, which is a bit bigger than the last calculation, but we can totally handle it. Remember, taking it one step at a time is the key.

So, let’s start with the multiplication: 2017 * 2017. When you multiply these two, you get 4068289. That's a pretty big number, but don't let it intimidate you! Now we can rewrite our expression as 2017 + 4068289 + 2018. See how we're making progress? We've knocked out the multiplication, and now we just have addition left to do.

Now we need to add 2017, 4068289, and 2018 together. When you add these three numbers, 2017 plus 4068289 plus 2018 equals 4072324. This is the final answer for the second expression. We took a potentially intimidating problem and broke it down into simpler steps. Remember, always prioritize multiplication before addition. So, for part (b), our final answer is 4072324. High five! Let's move on to the next one.

To recap:

  1. Multiplication: 2017 * 2017 = 4068289
  2. Addition: 2017 + 4068289 + 2018 = 4072324

Thus, 2017 + 2017 * 2017 + 2018 = 4072324.

c) Calculating 2001 * 25 + 75 * (5001 - 3000)

Okay, guys, let's tackle this next expression: 2001 * 25 + 75 * (5001 - 3000). This one looks a bit more complex because we've got parentheses in the mix. Remember PEMDAS/BODMAS? Parentheses come first! So, our very first step is to deal with what's inside those parentheses. We need to figure out what 5001 minus 3000 is.

Let's start with the parentheses: (5001 - 3000). When you subtract 3000 from 5001, you get 2001. Now we can rewrite the expression as 2001 * 25 + 75 * 2001. Great! We've simplified the parentheses, and the expression is looking a bit more manageable now. We're making good progress!

Next up, we need to handle the multiplication. We actually have two multiplication operations here: 2001 * 25 and 75 * 2001. Let's do them one at a time. First, 2001 multiplied by 25 equals 50025. And second, 75 multiplied by 2001 equals 150075. Now we can rewrite our expression as 50025 + 150075. We've knocked out the multiplications, and we're down to the final step!

Finally, let's add 50025 and 150075 together. When you add these two numbers, 50025 plus 150075 comes out to 200100. And that's our final answer for this expression! We successfully navigated the parentheses and multiplications, and landed on the solution. Great job, guys! Let's break down the steps we took:

  1. Parentheses: 5001 - 3000 = 2001
  2. Multiplication: 2001 * 25 = 50025
  3. Multiplication: 75 * 2001 = 150075
  4. Addition: 50025 + 150075 = 200100

Thus, 2001 * 25 + 75 * (5001 - 3000) = 200100.

d) Calculating 2017 + 2017 * 2018 - 2019 * 2016

Alright, let's dive into our last expression: 2017 + 2017 * 2018 - 2019 * 2016. This one looks like a doozy, but don't worry, we'll break it down just like the others. Remember, order of operations is our best friend! We've got multiplication and then addition and subtraction, so we'll tackle the multiplications first. Buckle up, here we go!

First up, we need to handle the multiplications. We've got two of them: 2017 * 2018 and 2019 * 2016. Let's take them one at a time. 2017 multiplied by 2018 equals 4070306. And 2019 multiplied by 2016 equals 4070304. Now we can rewrite our expression as 2017 + 4070306 - 4070304. We've conquered the multiplications, and the expression is looking much simpler.

Now we have addition and subtraction left. We do these from left to right. So first, we add 2017 and 4070306. That gives us 4072323. Now our expression is 4072323 - 4070304. We're in the home stretch!

Finally, we subtract 4070304 from 4072323. When you do that, you get 2019. And that's our final answer for this expression! We navigated the multiplications, addition, and subtraction, and landed on the solution. Awesome job, team! Let's recap the steps we took:

  1. Multiplication: 2017 * 2018 = 4070306
  2. Multiplication: 2019 * 2016 = 4070304
  3. Addition: 2017 + 4070306 = 4072323
  4. Subtraction: 4072323 - 4070304 = 2019

Thus, 2017 + 2017 * 2018 - 2019 * 2016 = 2019.

Conclusion

So there you have it, guys! We've successfully calculated all four expressions step by step. Remember, the key to tackling these problems is to follow the order of operations and break things down into smaller, manageable steps. You've got this! Keep practicing, and you'll become a math whiz in no time. Great job today, and let's keep crushing those calculations!