Solving For A - B: If 5√2=√a And 2√5=√b

Hey guys! Today, we're diving into a fun little math problem that involves square roots and some basic algebra. We're given two equations: 5√2 = √a and 2√5 = √b, and our mission, should we choose to accept it, is to find the value of a - b. Don't worry, it's not as intimidating as it looks! We'll break it down step by step, making sure everyone can follow along. Think of it like piecing together a puzzle – each step gets us closer to the final answer. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into calculations, let's make sure we understand what the problem is asking. We have two equations that relate square roots of numbers (√a and √b) to simpler expressions (5√2 and 2√5). Our ultimate goal is to find the difference between a and b. This means we first need to figure out what a and b actually are. The key here is to remember how to manipulate equations involving square roots. We'll be using the principle that if we square both sides of an equation, we maintain the equality. This will help us get rid of the square root signs and isolate a and b. It's like unwrapping a present – we need to carefully remove the outer layers to reveal what's inside. We'll also need to remember some basic arithmetic operations like squaring numbers and multiplying. This is where those math fundamentals we learned way back when really come in handy. So, are you ready to unwrap this mathematical present with me? Let's move on to the first step: finding the value of a.

Finding the Value of a

Okay, let's tackle the first part of our mission: finding the value of a. We're given the equation 5√2 = √a. Remember our trick for getting rid of square roots? That's right, we're going to square both sides of the equation. This is a crucial step, so let's take it slow and make sure we understand each part. Squaring the left side, (5√2)², means we're multiplying (5√2) by itself. We can think of this as (5√2) * (5√2). Now, remember the rules of multiplying radicals: we multiply the numbers outside the square roots (the 5s) and the numbers inside the square roots (the 2s). So, 5 * 5 = 25, and √2 * √2 = 2. Therefore, (5√2)² becomes 25 * 2, which equals 50. On the right side, squaring √a simply gives us a. Why? Because the square root and the square operations are inverses of each other – they cancel each other out. So, after squaring both sides, our equation becomes 50 = a. Voila! We've found the value of a. Now that wasn't so bad, was it? We've successfully unwrapped the first part of our mathematical present. But our journey isn't over yet – we still need to find the value of b. Let's head on to the next section and conquer that challenge!

Finding the Value of b

Alright, team, we've successfully found the value of a, and now it's time to set our sights on b. We're given the equation 2√5 = √b. Just like before, our trusty method of squaring both sides will come to the rescue. This will help us eliminate the square root and isolate b. So, let's square both sides of the equation. On the left side, we have (2√5)². Remember how we tackled this before? We're multiplying (2√5) by itself, which is (2√5) * (2√5). Again, we multiply the numbers outside the square root and the numbers inside. So, 2 * 2 = 4, and √5 * √5 = 5. Therefore, (2√5)² becomes 4 * 5, which equals 20. On the right side, squaring √b simply gives us b, just like before. The square and the square root cancel each other out, leaving us with the variable we're trying to find. So, after squaring both sides, our equation transforms into 20 = b. Fantastic! We've successfully unearthed the value of b. Now we're really cooking! We've got both a and b in our mathematical toolbox. The final piece of the puzzle is within reach – we just need to find the difference between them.

Calculating a - b

Okay, folks, we're in the home stretch! We've found that a equals 50 and b equals 20. The problem asks us to find the value of a - b. This is a straightforward subtraction problem now that we know the values of a and b. So, we simply substitute the values we found into the expression: a - b = 50 - 20. Now, let's do the math. 50 minus 20 is, drumroll please… 30! So, a - b = 30. And there you have it! We've successfully navigated the problem and arrived at our final answer. We started with equations involving square roots, squared both sides to eliminate the radicals, found the individual values of a and b, and then subtracted to find the difference. It's like climbing a mountain – we took it one step at a time, and now we're standing at the summit, victorious!

Final Answer

So, to recap, if 5√2 = √a and 2√5 = √b, then a - b = 30. We've successfully solved the problem! This involved squaring both sides of the equations to eliminate the square roots, calculating the values of 'a' and 'b' individually, and then performing a simple subtraction. Great job, everyone! You've tackled this mathematical challenge with skill and perseverance. Remember, the key to solving problems like this is to break them down into smaller, manageable steps. And don't be afraid to use the tools in your mathematical toolbox, like squaring both sides of an equation, to simplify things. Keep practicing, and you'll become a math whiz in no time! This problem highlights the importance of understanding how square roots and algebraic manipulations work. These are fundamental concepts in mathematics, and mastering them will help you tackle more complex problems down the road. So, keep up the great work, and keep exploring the fascinating world of math!