Square Root Calculations: Solve √16, √3.24, And More

Hey guys! Today, we're diving into the fascinating world of square roots and tackling some calculations. This might seem tricky at first, but trust me, once you get the hang of it, it's super straightforward. We'll break down each part step by step, making sure you understand exactly what's going on. So, let’s jump right in and solve these square root problems together!

a. √16

Let's start with the basics. Our first task is to calculate the square root of 16, written as √16. When we talk about square roots, we're asking ourselves: what number, when multiplied by itself, gives us 16? This is a fundamental concept in mathematics, and understanding it is crucial for more complex calculations down the road. Think of it like finding the side length of a square when you know its area is 16 square units. So, what’s the magic number here?

The square root of 16 is a classic example that many people learn early in their math journey. It's one of those perfect squares that pops up frequently, making it essential to memorize. When you’re trying to figure this out, you might start by trying out a few numbers. For instance, you know that 2 times 2 is 4, which is too small. Then you might try 3 times 3, which equals 9, still too small. But when you try 4 times 4, you get 16! Bingo! So, 4 multiplied by itself gives us 16. This means that the square root of 16 is 4. We can write this as √16 = 4.

Understanding this basic square root is the foundation for tackling more complicated problems. It’s like learning the alphabet before you can read a book. This simple calculation is a building block for algebra, geometry, and even real-world applications like construction and engineering. So, mastering the basics like √16 is super important. You’ll see these kinds of calculations pop up again and again, so getting comfortable with them now will save you time and frustration later. Keep practicing, and you'll become a square root pro in no time!

b. √3.24

Next up, we're tackling the square root of 3.24, written as √3.24. Now, this one might look a bit more intimidating because we're dealing with a decimal number. But don't worry, the same basic principle applies: we need to find a number that, when multiplied by itself, equals 3.24. Sometimes, dealing with decimals can feel like navigating a maze, but we’ll break it down to make it super easy. Think of it as figuring out the side length of a square with an area of 3.24 square units. Ready to find that number?

When faced with a decimal like 3.24, it's helpful to think about perfect squares that are close to it. We know that 1 squared (1 * 1) is 1, and 2 squared (2 * 2) is 4. So, the square root of 3.24 must be somewhere between 1 and 2. This narrows down our possibilities and makes the problem more manageable. Next, we can try some decimals in that range. For example, we might try 1.5. When you multiply 1.5 by 1.5, you get 2.25, which is too small. So, we need to go higher. Let's try 1.8. If we multiply 1.8 by 1.8, we get 3.24! That's our answer!

Therefore, the square root of 3.24 is 1.8. We can write this as √3.24 = 1.8. Dealing with decimals might seem tricky at first, but with practice, you'll get the hang of it. Remember, you can always use a calculator to check your answer, but understanding the process is what's really important. Figuring out square roots like this helps build your problem-solving skills and gives you a solid foundation for more advanced math topics. Keep up the great work, guys!

c. √25 + √49 - √36

Alright, let's move on to something a bit more complex: √25 + √49 - √36. Here, we're not just dealing with one square root, but a combination of them! This means we need to calculate each square root individually and then perform the addition and subtraction. Don’t let it overwhelm you; just take it one step at a time. Think of it as solving a puzzle – each piece (square root) fits together to give us the final answer. So, let’s start by identifying each of those pieces.

First, let's tackle √25. We need to find a number that, when multiplied by itself, equals 25. Think about your times tables – what number times itself gives you 25? That’s right, it’s 5! So, √25 = 5.

Next, we have √49. Again, we're looking for a number that, when multiplied by itself, equals 49. If you know your squares, you'll remember that 7 times 7 is 49. So, √49 = 7.

Finally, let's calculate √36. What number multiplied by itself gives us 36? If you guessed 6, you're spot on! So, √36 = 6.

Now that we've found the value of each square root, we can plug them back into the original equation: √25 + √49 - √36 becomes 5 + 7 - 6. Now it's just simple addition and subtraction. 5 plus 7 equals 12, and then 12 minus 6 equals 6. So, the answer is 6! We can write this out as √25 + √49 - √36 = 5 + 7 - 6 = 6. Breaking down the problem into smaller parts made it much easier to solve, didn't it? Great job, everyone!

d. √(6² + 8²) ÷ √(4 * 5²)

Last but definitely not least, let's tackle this final challenge: √(6² + 8²) ÷ √(4 * 5²). This looks like the most complex one yet, but don't sweat it! We’re going to use the same step-by-step approach that worked so well before. Remember, the key is to break it down into smaller, manageable parts. We’ve got squares, square roots, multiplication, addition, and division all in one equation. It’s like a mathematical adventure! Are you ready to embark on it?

First, let's simplify the expressions inside the square roots. We’ll start with the first square root: √(6² + 8²). We need to calculate 6 squared (6²) and 8 squared (8²) before we can add them. 6 squared is 6 times 6, which equals 36. 8 squared is 8 times 8, which equals 64. Now we add these together: 36 + 64 = 100. So, the first square root becomes √100.

Next, let's simplify the second square root: √(4 * 5²). We first need to calculate 5 squared (5²), which is 5 times 5, or 25. Then we multiply this by 4: 4 * 25 = 100. So, the second square root also becomes √100.

Now our equation looks much simpler: √100 ÷ √100. We need to find the square root of 100. What number multiplied by itself gives us 100? If you thought of 10, you're absolutely right! So, √100 = 10.

Finally, we divide: 10 ÷ 10 = 1. Therefore, the answer is 1! We can write this all out as √(6² + 8²) ÷ √(4 * 5²) = √(36 + 64) ÷ √(4 * 25) = √100 ÷ √100 = 10 ÷ 10 = 1. See? Even the trickiest-looking problems become manageable when you break them down step by step. You guys are math superstars!

Conclusion

So there you have it! We've successfully calculated √16, √3.24, √25 + √49 - √36, and √(6² + 8²) ÷ √(4 * 5²). Remember, the key to mastering square roots is practice and breaking down problems into smaller steps. You've done an amazing job following along, and I'm super proud of your hard work. Keep practicing, and you'll become a square root whiz in no time! You've got this!