Simplify √(2/3) - √(3/2): A Step-by-Step Solution

Hey guys! Ever stumbled upon a math problem that looks like it's speaking a different language? Well, today we're tackling one of those head-scratchers: simplifying the expression √2/3 - √3/2. It might seem intimidating at first, but trust me, we'll break it down step-by-step until it's as clear as day. So, grab your thinking caps and let's dive into the fascinating world of square roots and fractions!

Understanding the Challenge: The Square Root Expression

Before we jump into solving, let's really understand what this expression is asking us. We're dealing with the difference between two square roots, each containing a fraction. √2/3 means the square root of the fraction two-thirds, and √3/2 means the square root of three-halves. Our goal is to simplify this difference into a more manageable form. This involves a few key mathematical concepts like rationalizing denominators, finding common denominators, and simplifying radicals. It's like a puzzle with multiple pieces, and we're going to fit them all together. Remember, math isn't just about getting the right answer; it's about the journey of understanding the how and why behind the solution. So, let's get started on this mathematical adventure!

Breaking Down the Square Roots

The initial step in simplifying √2/3 - √3/2 is to tackle each square root individually. We can rewrite √2/3 as √2 / √3 and √3/2 as √3 / √2. This separation is crucial because it allows us to work with the numerator and denominator separately. Now, here's a little secret weapon in our math arsenal: rationalizing the denominator. This technique helps us eliminate square roots from the denominator, making the expression easier to work with. To rationalize the denominator of √2 / √3, we multiply both the numerator and the denominator by √3. Similarly, for √3 / √2, we multiply both the numerator and the denominator by √2. This might seem like a small step, but it's a game-changer in simplifying our expression. Think of it as clearing the path for the rest of our solution. It's like decluttering your workspace before starting a big project – it makes everything flow smoother!

Rationalizing the Denominators: A Key Technique

Let's put that rationalizing skill into action! When we multiply √2 / √3 by √3 / √3, we get (√2 * √3) / (√3 * √3), which simplifies to √6 / 3. Similarly, multiplying √3 / √2 by √2 / √2 gives us (√3 * √2) / (√2 * √2), which simplifies to √6 / 2. See how we've cleverly eliminated the square roots from the denominators? This is the magic of rationalizing! Now our expression looks much friendlier: √6 / 3 - √6 / 2. We've transformed the problem from something that looked complex to a more approachable form. It's like turning a tangled knot into a neat loop. This step is a testament to the power of mathematical techniques in simplifying problems. Remember, each step we take is a step closer to the final answer. So, let's keep going!

Finding a Common Denominator: Combining the Fractions

Now that we have √6 / 3 - √6 / 2, the next challenge is to subtract these fractions. And as you guys probably remember from your fraction-fighting days, we need a common denominator to do that! The least common multiple of 3 and 2 is 6, so that's our target. To get a denominator of 6 for the first fraction (√6 / 3), we multiply both the numerator and denominator by 2, resulting in 2√6 / 6. For the second fraction (√6 / 2), we multiply both the numerator and denominator by 3, giving us 3√6 / 6. Now our expression looks like this: 2√6 / 6 - 3√6 / 6. We've successfully brought the fractions under a common denominator, which means we're ready for the final subtraction. This step is like aligning the pieces of a puzzle so they can fit together perfectly. It's all about preparing the ground for the final solution.

The Final Subtraction and Simplification

With a common denominator in place, we can finally perform the subtraction: 2√6 / 6 - 3√6 / 6. This is like combining like terms in algebra – we're essentially subtracting the coefficients of √6. So, 2√6 minus 3√6 is simply -√6. This gives us -√6 / 6. And there you have it! We've successfully simplified the original expression to -√6 / 6. This is our final answer, a neat and simplified form of the initial expression. It's like reaching the summit of a mountain after a challenging climb. We started with a complex-looking expression and, through careful steps and techniques, arrived at a clear and concise solution. This journey highlights the beauty of mathematics – the ability to transform complexity into simplicity.

In Conclusion: Mastering Square Root Simplification

So, guys, we've tackled the expression √2/3 - √3/2 head-on and emerged victorious! We've seen how breaking down the problem into smaller steps, rationalizing denominators, finding common denominators, and simplifying radicals can lead us to the solution. The final simplified form is -√6 / 6. But more than just getting the right answer, we've explored the underlying concepts and techniques that make it possible. This is the true essence of learning mathematics – not just memorizing formulas, but understanding the why behind the how. So, the next time you encounter a similar challenge, remember the steps we've taken today, and you'll be well-equipped to conquer it. Keep practicing, keep exploring, and most importantly, keep enjoying the fascinating world of mathematics! You've got this!