Solving: 4(5² + 20 ÷ 5) - A Step-by-Step Guide

Hey guys! Today, let's break down how to solve the mathematical expression $4(5^2 + 20 extdiv 5)$. It might look a bit intimidating at first, but don't worry! We’ll go through it together step by step, making sure everyone understands the process. We'll cover the order of operations, which is super important in math, and show you exactly how to tackle this kind of problem. By the end of this guide, you'll not only know the answer but also understand the why behind each step. So, grab your pencils and let’s dive in! Whether you're a student tackling homework or just someone who enjoys a good math challenge, this breakdown will help clarify the process. Remember, math is like building blocks – once you understand the basics, more complex problems become much easier to handle. We'll focus on clarity and making sure you grasp the underlying principles so you can apply them to similar problems in the future.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we jump into solving the expression, let's quickly recap the order of operations, often remembered by the acronyms PEMDAS or BODMAS. This order is crucial because it tells us which operations to perform first to get the correct answer. If you mess up the order, you’ll end up with the wrong result, and nobody wants that, right? Think of it like following a recipe – you need to add the ingredients in the right sequence to bake a perfect cake. Same thing with math!

• Parentheses (or Brackets): First, we deal with anything inside parentheses or brackets.
• Exponents (or Orders): Next up are exponents, like squares and cubes.
• Multiplication and Division: Then, we handle multiplication and division from left to right.
• Addition and Subtraction: Finally, we do addition and subtraction, also from left to right.

Why is this order so important? Imagine if we didn't have a set order. Different people could solve the same expression and come up with different answers! This would lead to chaos in mathematics, and we definitely don’t want that. The order of operations ensures that everyone solves the problem in the same way, guaranteeing a consistent and correct solution. It's like having a universal language for math, making sure we're all on the same page.

Let's make this even clearer with a simple example. Suppose we have the expression $2 + 3 imes 4$. If we just went from left to right, we might think the answer is $5 imes 4 = 20$. But, following PEMDAS/BODMAS, we do the multiplication first: $3 imes 4 = 12$, and then add 2: $2 + 12 = 14$. See the difference? That's why understanding and applying the order of operations is absolutely vital. So, keep PEMDAS/BODMAS in mind as we tackle our main problem – it’s our guiding star in the world of mathematical expressions!

Breaking Down the Expression: $4(5^2 + 20 extdiv 5)$

Okay, now that we've refreshed our memory on the order of operations, let's dive into our expression: $4(5^2 + 20 extdiv 5)$. The best way to tackle a problem like this is to break it down into smaller, more manageable chunks. Think of it like eating an elephant – you wouldn't try to swallow it whole, right? You'd take it one bite at a time. Same principle here! We're going to dissect this expression piece by piece, making sure we don't miss any crucial details.

First up, we notice those parentheses. Remember from PEMDAS/BODMAS that anything inside parentheses gets our attention first. Inside the parentheses, we have two operations: an exponent ($5^2$) and division ($20 extdiv 5$). According to our order of operations, we need to deal with the exponent before the division. So, let’s start there. Exponents tell us to multiply a number by itself a certain number of times. In this case, $5^2$ means 5 multiplied by itself, which is $5 imes 5 = 25$. Easy peasy!

Now we can rewrite our expression, replacing $5^2$ with 25: $4(25 + 20 extdiv 5)$. See how we’re making progress? Next, we still have division inside the parentheses. We have $20 extdiv 5$, which means 20 divided by 5. If you know your times tables, you'll know that 20 divided by 5 is 4. So, we replace $20 extdiv 5$ with 4, and our expression becomes $4(25 + 4)$. We're getting closer to the finish line!

We’re still inside the parentheses, so we need to finish up there before we move on. We now have a simple addition: $25 + 4$. Adding those together gives us 29. Our expression is now looking much simpler: $4(29)$. The parentheses are still there, but now they just mean multiplication. We’ve simplified everything inside them, so we can move on to the next step. By breaking the expression down like this, we’ve turned a potentially confusing problem into a series of simple steps. Remember, that’s the key to mastering math – taking things one step at a time.

Solving the Expression Step-by-Step

Alright, we've laid the groundwork, understood the order of operations, and broken down our expression. Now, let's put it all together and solve $4(5^2 + 20 extdiv 5)$ step-by-step. This is where we get to see all our hard work pay off, and it's super satisfying when the pieces fall into place. Trust me, math can be like solving a puzzle – each step gets you closer to the final picture!

1. Address the Exponent: First, we tackle the exponent inside the parentheses: $5^2 = 5 imes 5 = 25$. We're following PEMDAS/BODMAS, so exponents come before division or addition. This gives us $4(25 + 20 extdiv 5)$.
2. Perform the Division: Next up, we handle the division inside the parentheses: $20 extdiv 5 = 4$. So, our expression becomes $4(25 + 4)$. We're simplifying bit by bit, and it's looking good!
3. Complete the Addition Inside Parentheses: Now, we add the numbers inside the parentheses: $25 + 4 = 29$. Our expression is now $4(29)$. We've successfully simplified everything inside the parentheses – woohoo!
4. Multiply: Finally, we perform the multiplication: $4 imes 29$. This is the last step, and it's a straightforward multiplication problem. If you need to, you can break it down further: $4 imes 20 = 80$ and $4 imes 9 = 36$. Adding those together, $80 + 36 = 116$.

So, the final answer is 116! See? We got there by following the order of operations and breaking the problem into manageable steps. It might seem like a lot of steps, but each one is simple on its own. That’s the beauty of math – complex problems can be solved by breaking them down into smaller, easier parts. By going through this step-by-step process, we not only found the solution but also reinforced our understanding of the underlying principles. Remember, it's not just about getting the answer; it's about understanding how we got there. That’s what really makes the difference in mastering math.

The Final Answer and Key Takeaways

Drumroll, please! We've reached the end of our mathematical journey, and the final answer to the expression $4(5^2 + 20 extdiv 5)$ is f{116}. Give yourself a pat on the back – you did it! We started with a seemingly complex expression and, by breaking it down and following the order of operations, we arrived at a clear and precise solution. But it's not just about the answer; it’s about the journey and what we learned along the way. So, let's recap some key takeaways from this exercise.

First and foremost, the order of operations (PEMDAS/BODMAS) is your best friend in math. It’s the guiding principle that ensures we solve expressions correctly every time. Remember the acronyms: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction. Follow this order, and you’ll avoid common pitfalls and arrive at the right answer. It's like having a roadmap – it keeps you on the right path!

Another crucial takeaway is the importance of breaking down complex problems. Math can sometimes feel overwhelming, especially when you're faced with long or complicated expressions. But, just like we did with our expression, breaking it down into smaller, more manageable parts makes the whole process much less daunting. Each step becomes simpler, and you can focus on one operation at a time. This approach not only makes the problem easier to solve but also helps you understand the logic behind each step.

Finally, remember that practice makes perfect. The more you work with mathematical expressions, the more comfortable and confident you’ll become. Don't be afraid to tackle different types of problems, and don't get discouraged if you make mistakes along the way. Mistakes are a natural part of learning, and they provide valuable opportunities for growth. Each time you work through a problem, you're reinforcing your understanding and building your skills. So, keep practicing, keep learning, and most importantly, keep having fun with math! We've conquered this expression together, and now you're better equipped to tackle whatever mathematical challenges come your way.