Solving (1 + 4 + √4) / (125 + √16) A Step-by-Step Guide

Hey guys! Today, we're diving into a fun little math problem. We're going to break down how to calculate the expression (1 + 4 + √4) / (125 + √16). Don't worry, it's not as scary as it looks! We'll go through each step together, so you'll be a pro in no time. So, let's get started and make math a bit more fun, shall we?

Understanding the Problem

Before we jump into the calculation, let's make sure we understand exactly what we're dealing with. The problem asks us to evaluate the expression (1 + 4 + √4) / (125 + √16). This means we need to simplify both the numerator (the top part of the fraction) and the denominator (the bottom part) and then divide the result. Remember, the key to solving any math problem is to break it down into smaller, manageable steps. So, that's exactly what we're going to do.

First off, let's talk about the different parts of this expression. We've got some simple addition, which is straightforward. But we also have square roots (√), which might seem a bit intimidating if you're not super familiar with them. A square root of a number is a value that, when multiplied by itself, gives you the original number. For example, the square root of 4 (√4) is 2 because 2 * 2 = 4. Similarly, the square root of 16 (√16) is 4 because 4 * 4 = 16. Knowing this, we can start simplifying the expression.

When you're faced with an expression like this, it's crucial to follow the order of operations, often remembered by the acronym PEMDAS (or BODMAS in some regions). This stands for:

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

In our case, we have square roots, which fall under the 'Exponents' category, and then we have addition and division. So, we'll tackle the square roots first, then the addition in both the numerator and the denominator, and finally, we'll perform the division. By following this order, we ensure we get the correct answer. Missteps in the order of operations can easily lead to the wrong result, so it’s a really important thing to keep in mind. Alright, now that we've got a solid plan, let's get into the nitty-gritty of solving this problem. Let’s dive in and start simplifying!

Simplifying the Numerator (1 + 4 + √4)

Okay, let's kick things off by simplifying the numerator of our expression, which is (1 + 4 + √4). The numerator, as a reminder, is the top part of the fraction. To simplify this, we'll follow the order of operations (PEMDAS/BODMAS), but in this case, it's pretty straightforward. We have addition and a square root, so we'll handle the square root first. Remember, breaking down the problem into smaller steps is the name of the game here. This makes it less intimidating and easier to manage.

So, let's tackle that square root: √4. What number, when multiplied by itself, gives us 4? That's right, it's 2! So, we can replace √4 with 2 in our expression. Now, our numerator looks like this: 1 + 4 + 2. See? We're already making progress!

Now that we've dealt with the square root, we're left with simple addition. We just need to add the numbers together. We have 1 + 4 + 2. Let’s add them up: 1 plus 4 is 5, and then 5 plus 2 is 7. So, the simplified numerator is 7. Great job, guys! We've successfully simplified the top part of our fraction.

It's always a good idea to double-check your work, especially with simple calculations like this. Make sure you didn't make any silly mistakes. We replaced √4 with 2, and then we added 1 + 4 + 2, which indeed equals 7. So, we're confident in our result.

Simplifying the numerator is a crucial step in solving the overall problem. By breaking it down and tackling the square root first, we made the addition part super easy. Now that we have the simplified numerator, we can move on to the denominator. We'll follow the same approach, simplifying it step by step. Keep up the great work, and let's get that denominator sorted out!

Simplifying the Denominator (125 + √16)

Alright, now that we've conquered the numerator, let's turn our attention to the denominator, which is (125 + √16). Just like with the numerator, we're going to break this down step by step to make it super manageable. The denominator is the bottom part of our fraction, and we need to simplify it before we can do the final division. So, let’s get to it!

Looking at the expression (125 + √16), we see that we have addition and a square root. Following the order of operations (PEMDAS/BODMAS), we need to tackle the square root first. So, let's focus on √16. What number, when multiplied by itself, gives us 16? Think about it for a second... That's right, it's 4! Because 4 * 4 = 16. So, we can replace √16 with 4 in our expression. Now, our denominator looks like this: 125 + 4. We're making good progress, guys!

Now that we've dealt with the square root, we're left with a simple addition problem. We just need to add 125 and 4 together. This is pretty straightforward: 125 plus 4 equals 129. So, the simplified denominator is 129. Awesome job! We've successfully simplified the bottom part of our fraction.

Before we move on, let's double-check our work, just to be sure. We replaced √16 with 4, and then we added 125 + 4, which indeed equals 129. So, we're confident in our result. It’s always a good practice to double-check, especially with these smaller steps, to avoid any errors that could throw off the final answer.

Simplifying the denominator is a crucial step, just like simplifying the numerator. By breaking it down and handling the square root first, we made the addition part nice and easy. Now that we have both the simplified numerator and the simplified denominator, we're ready to move on to the final step: dividing the numerator by the denominator. We're almost there, keep up the fantastic work!

Dividing the Simplified Numerator by the Simplified Denominator

Okay, guys, we've reached the final stage of our calculation! We've successfully simplified both the numerator and the denominator. Remember, we found that the simplified numerator is 7, and the simplified denominator is 129. Now, the last step is to divide the numerator by the denominator. This means we need to calculate 7 / 129. So, let's dive in and get this done!

We have the fraction 7 / 129. This fraction represents the result of our original expression. At this point, we need to determine if this fraction can be simplified further. To simplify a fraction, we look for common factors between the numerator and the denominator. A common factor is a number that divides both the numerator and the denominator evenly.

Let's think about the factors of 7. Since 7 is a prime number, its only factors are 1 and 7. Now, we need to check if 129 is divisible by 7. To do this, we can perform the division: 129 ÷ 7. If we do the division, we find that 129 is not divisible by 7 without leaving a remainder. This means that 7 and 129 do not have any common factors other than 1. When the only common factor is 1, the fraction is in its simplest form.

Therefore, the fraction 7 / 129 is already in its simplest form. This is our final answer! We have successfully calculated the value of the expression (1 + 4 + √4) / (125 + √16).

So, to recap, we started with a seemingly complex expression, but by breaking it down into smaller, manageable steps, we were able to solve it. We first simplified the numerator, then the denominator, and finally, we divided the simplified numerator by the simplified denominator. We also checked if our final fraction could be simplified further, which it couldn't. This is a great example of how breaking down a problem and taking it one step at a time can make even the trickiest-looking questions solvable.

Final Answer and Conclusion

Alright, let's wrap things up! We've journeyed through the entire process of calculating the expression (1 + 4 + √4) / (125 + √16), and we've reached our final answer. To quickly recap, we simplified the numerator (1 + 4 + √4) to 7, and we simplified the denominator (125 + √16) to 129. Then, we divided the simplified numerator by the simplified denominator, giving us the fraction 7 / 129.

We also determined that the fraction 7 / 129 is already in its simplest form because 7 and 129 do not share any common factors other than 1. Therefore, our final answer is 7 / 129. Woohoo! We did it, guys!

So, the solution to the expression (1 + 4 + √4) / (125 + √16) is 7 / 129. This whole process highlights the importance of following the order of operations (PEMDAS/BODMAS) and breaking down complex problems into simpler steps. By tackling each part of the expression individually, we made the entire calculation much more manageable.

Remember, guys, math might seem intimidating at first, but with a systematic approach and a bit of practice, you can conquer any problem. Breaking things down, double-checking your work, and understanding the fundamental rules are key to success. This problem was a great example of that. We handled square roots, addition, and division, all in one go! You should be proud of yourselves for sticking with it and working through the solution. Keep up the great work, and remember, every math problem is just a puzzle waiting to be solved. And who doesn't love a good puzzle? Keep practicing, keep learning, and you'll be a math whiz in no time!

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