Simplify 1/√(x-1) + √(x-1): A Step-by-Step Guide
Introduction
Hey guys! Let's dive into simplifying a cool little mathematical expression today. We're going to tackle: $\frac{1}{\sqrt{x-1}} + \sqrt{x-1}$. This expression looks a bit tangled at first glance, but don't worry, we'll break it down step by step and make it super easy to understand. Our main goal here is to combine these terms into a single, cleaner fraction. This kind of simplification is super useful in calculus, algebra, and many other areas of math. Think of it as decluttering your mathematical workspace – a tidy expression is much easier to work with! We'll be using some basic algebraic techniques, such as finding a common denominator and simplifying radicals. These are fundamental skills that will help you tackle more complex problems later on. So, grab your thinking caps, and let's get started on this mathematical adventure! We'll explore why such simplifications are important, where you might encounter similar problems, and how mastering these skills can boost your math confidence. Remember, every complex problem is just a series of simple steps, and we're here to guide you through each one.
Understanding the Components
Before we jump into the simplification process, let's take a moment to understand the different parts of our expression: $\frac{1}{\sqrt{x-1}} + \sqrt{x-1}$. First, we have a fraction, $\frac{1}{\sqrt{x-1}}$, where the denominator involves a square root. Square roots can sometimes make things a bit tricky, especially when they're in the denominator of a fraction. Then, we have $&sqrt{x-1}$, which is a simple square root term. Notice that both terms involve $&sqrt{x-1}$, which gives us a hint that we can probably combine them somehow. The expression inside the square root, x - 1, is also important. We need to remember that whatever value x takes, x - 1 must be greater than zero. Why? Because we can't take the square root of a negative number (at least, not in the realm of real numbers), and we also can't have zero in the denominator of a fraction. This means x must be greater than 1. Keeping track of these little details is crucial in mathematics. They help us avoid common mistakes and ensure our solutions make sense. Understanding the components also sets the stage for choosing the right simplification strategy. In this case, recognizing the common term $&sqrt{x-1}$ suggests that finding a common denominator is the way to go. So, with a clear understanding of the pieces, let's move on to the exciting part: simplifying the expression!
Finding a Common Denominator
Alright, guys, let’s get our hands dirty and start simplifying! The first key step here is to find a common denominator for the two terms in our expression: $\frac1}{\sqrt{x-1}} + \sqrt{x-1}$. Currently, we have one term with a denominator of $&sqrt{x-1}$ and another term that, technically, has a denominator of 1 (since any number divided by 1 is itself). To combine these terms, we need them to have the same denominator. So, what's the common denominator we should use? You guessed it$. To get the second term, $&sqrtx-1}$, to have this denominator, we need to multiply it by $&frac{\sqrt{x-1}}{\sqrt{x-1}}$. Remember, multiplying by a fraction that's equal to 1 doesn't change the value of the term, just its appearance. So, let's do it \times \frac{\sqrt{x-1}}{\sqrt{x-1}} = \frac{(\sqrt{x-1})^2}{\sqrt{x-1}}$. Now, we know that squaring a square root cancels it out, so $(\sqrt{x-1})^2$ simplifies to x - 1. This gives us $&frac{x-1}{\sqrt{x-1}}$. Now that both terms have the same denominator, we can combine them. This is a super important step because it transforms our expression from a sum of two fractions into a single fraction, which is much easier to manage. So, we're making progress! Let's see what happens when we actually combine these fractions.
Combining the Fractions
Now that we've got a common denominator, the fun really begins! We can now combine our two fractions. Remember, we have: $\frac1}{\sqrt{x-1}} + \frac{x-1}{\sqrt{x-1}}$. Since they both have the same denominator, $&sqrt{x-1}$, we can simply add the numerators. This gives us\sqrt{x-1}}$. Look closely at the numerator{\sqrt{x-1}}$. Awesome! We've made some significant progress. We started with a sum of two terms and now we have a single fraction. But, we're not quite done yet. In mathematics, it's generally considered good practice to rationalize the denominator, which means getting rid of any square roots in the bottom of the fraction. This makes the expression cleaner and easier to work with in future calculations. So, our next step is to rationalize that denominator. Think of it as polishing our mathematical gem to make it shine even brighter! Are you ready to tackle the last piece of the puzzle?
Rationalizing the Denominator
Okay, guys, let's put the finishing touches on our simplified expression. We're currently sitting at $&fracx}{\sqrt{x-1}}$, and our mission is to get rid of that pesky square root in the denominator. This process is called rationalizing the denominator, and it's a common technique in algebra. So, how do we do it? The trick is to multiply both the numerator and the denominator by the square root that's causing the trouble. In our case, that's $&sqrt{x-1}$. Just like before, we're essentially multiplying by 1 (since $\frac{\sqrt{x-1}}{\sqrt{x-1}} = 1$), so we're not changing the value of the expression. Let's do it\sqrt{x-1}} \times \frac{\sqrt{x-1}}{\sqrt{x-1}} = \frac{x\sqrt{x-1}}{(\sqrt{x-1})^2}$. Now, remember that squaring a square root cancels it out, so $(\sqrt{x-1})^2$ simplifies to x - 1. This gives us}{x-1}$. And there you have it! We've successfully rationalized the denominator. Our final simplified expression is $&frac{x\sqrt{x-1}}{x-1}$. This looks much cleaner, doesn't it? It's easier to understand and work with. We started with a more complex expression and, through a series of clear, logical steps, we've arrived at a much simpler form. Give yourselves a pat on the back! You've just conquered a classic algebra problem.
Final Simplified Expression
So, after all our hard work, we've arrived at the final simplified expression: $\frac{x\sqrt{x-1}}{x-1}$. This is the result of taking our original expression, $&frac{1}{\sqrt{x-1}} + \sqrt{x-1}$, and applying a series of algebraic techniques to make it as clean and simple as possible. We found a common denominator, combined the fractions, and then rationalized the denominator to get rid of the square root in the bottom. This final form is not only more aesthetically pleasing to mathematicians (yes, we have a sense of aesthetics!), but it's also much more practical for further calculations. Imagine trying to plug the original expression into a calculus problem – it would be a bit of a headache. But this simplified form? Smooth sailing! Simplifying expressions like this is a fundamental skill in mathematics. It's like learning to organize your tools before starting a project – it makes everything else much easier. Plus, the process we've used here, finding common denominators and rationalizing, are techniques that pop up again and again in algebra, calculus, and beyond. So, by mastering these skills, you're not just solving one problem; you're building a foundation for future success in math. And that's something to be proud of!
Conclusion
Alright, guys, we've reached the end of our journey to simplify the expression $ rac1}{\sqrt{x-1}} + \sqrt{x-1}$, and what a journey it's been! We started with a slightly intimidating expression, broke it down into manageable parts, and then used our algebraic skills to combine and simplify it. We found a common denominator, added the fractions, and then rationalized the denominator to arrive at our final, elegant solution}{x-1}$. This whole process highlights the power of simplification in mathematics. It's not just about getting to the right answer; it's about making the problem easier to understand and work with. A simplified expression is like a well-organized map – it helps you see the path more clearly. The techniques we've used here are not just one-off tricks; they're fundamental tools that you'll use again and again in your mathematical adventures. So, the next time you encounter a complex expression, remember the steps we've taken today. Break it down, find common denominators, rationalize when necessary, and don't be afraid to get your hands dirty with the algebra. Math is like a puzzle, and simplification is one of the most satisfying ways to fit the pieces together. Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!