Simplifying $\frac{\sqrt{3}}{\sqrt{3}-\sqrt{x}}$: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into an exciting algebraic simplification problem. Our mission, should we choose to accept it, is to express the fraction $\frac{\sqrt{3}}{\sqrt{3}-\sqrt{x}}$ in its simplest form. This involves a bit of algebraic maneuvering, specifically a technique called rationalizing the denominator. So, grab your thinking caps, and let's embark on this mathematical journey together!

Rationalizing the Denominator: The Key to Simplicity

The main idea behind simplifying expressions like this is to get rid of the square root in the denominator. We don't like having those pesky radicals down there, as they make the expression look clunky and less user-friendly. The magic trick we use is called rationalizing the denominator. This technique involves multiplying both the numerator and the denominator by the conjugate of the denominator. But what exactly is a conjugate, you might ask? Well, the conjugate of an expression in the form a - b is simply a + b, and vice versa. It's like its mathematical twin, but with the opposite sign in the middle.

In our case, the denominator is $\sqrt{3} - \sqrt{x}$. So, its conjugate is $\sqrt{3} + \sqrt{x}$. Now, we're going to multiply both the numerator and the denominator of our fraction by this conjugate. Remember, multiplying by a fraction that's equal to 1 (like something divided by itself) doesn't change the value of the original expression, only its appearance. This is a crucial concept in algebra, guys, and it's what allows us to perform these transformations without altering the fundamental value of our expression. We're just giving it a makeover, a mathematical facelift if you will. We multiply both the top and bottom by the conjugate $\sqrt{3} + \sqrt{x}$ because this clever step eliminates the radical in the denominator. When we multiply the denominator by its conjugate, we leverage a special algebraic identity: (a - b)(a + b) = a² - b². This identity is a powerhouse in simplifying radical expressions, as it transforms a binomial involving square roots into a difference of squares, effectively banishing the radicals from the denominator. It's like waving a magic wand and making those square roots disappear!

So, let's perform this multiplication: $\frac{\sqrt{3}}{\sqrt{3}-\sqrt{x}} * \frac{\sqrt{3}+\sqrt{x}}{\sqrt{3}+\sqrt{x}}$. This step sets the stage for simplification, transforming the original fraction into a form where the denominator no longer contains a square root. By multiplying by the conjugate, we're essentially setting up a scenario where the difference of squares identity can work its magic, leading us closer to the simplest form of the expression. Think of it as a carefully choreographed dance, where each step is designed to bring us closer to our goal of a simplified expression. The multiplication ensures that the radicals in the denominator will cancel out, leaving us with a rational denominator and a more manageable expression.

Multiplying and Simplifying: Step-by-Step

Now that we've set up the multiplication, let's dive into the nitty-gritty details of actually performing it. We need to multiply the numerators together and the denominators together. For the numerator, we have $\sqrt{3} * (\sqrt{3} + \sqrt{x})$. This requires us to distribute the $\sqrt{3}$ across both terms inside the parentheses. Remember your distributive property, folks! It's a fundamental concept in algebra, and it's going to be our best friend here. When we distribute, we get $\sqrt{3} * \sqrt{3} + \sqrt{3} * \sqrt{x}$, which simplifies to 3 + $\sqrt{3x}$. The product $\sqrt{3} * \sqrt{3}$ is simply 3 because the square root of a number multiplied by itself is just the original number. And the product $\sqrt{3} * \sqrt{x}$ can be written as $\sqrt{3x}$ using the property that $\sqrt{a} * \sqrt{b} = \sqrt{ab}$. So, the numerator is shaping up nicely, and we're one step closer to simplifying the entire expression.

Moving on to the denominator, we have $(\sqrt{3} - \sqrt{x}) * (\sqrt{3} + \sqrt{x})$. This is where the magic of the conjugate comes into play! As we mentioned earlier, this is in the form (a - b)(a + b), which we know simplifies to a² - b². In our case, a is $\sqrt{3}$ and b is $\sqrt{x}$. So, the denominator becomes ($\sqrt{3}$)² - ($\sqrt{x}$)², which simplifies to 3 - x. Notice how the square roots have vanished from the denominator! This is the whole point of rationalizing the denominator, guys. We've successfully transformed the denominator into an expression without any square roots, making it much cleaner and easier to work with.

Putting it all together, our fraction now looks like this: $\frac{3 + \sqrt{3x}}{3 - x}$. This is a significant improvement over the original expression, as we've eliminated the square root from the denominator. The expression is now in a much more manageable form, and we can easily see the relationship between the numerator and the denominator. We've successfully navigated the algebraic terrain and arrived at a simplified expression. Give yourselves a pat on the back, folks! You've earned it. Remember, the key to simplifying these types of expressions is to identify the conjugate of the denominator and multiply both the numerator and the denominator by it. This technique allows us to eliminate the square root from the denominator and express the fraction in its simplest form.

Identifying the Simplest Form: The Final Answer

Now, let's compare our simplified expression, $\frac{3 + \sqrt{3x}}{3 - x}$, with the options provided. We can see that it matches one of the options exactly! This means we've successfully simplified the original expression and found its simplest form. It's like solving a puzzle, where each step brings you closer to the final solution. And in this case, the solution is the simplest form of the fraction, which we've identified by carefully applying algebraic techniques and simplifying the expression step-by-step. This final step solidifies our understanding of the simplification process and confirms that we've arrived at the correct answer. It's a moment of triumph, a culmination of our mathematical efforts. So, let's celebrate our success and move on to the next challenge, knowing that we have the skills and knowledge to tackle it head-on!

Therefore, the simplest form of $\frac{\sqrt{3}}{\sqrt{3}-\sqrt{x}}$ is $\frac{3+\sqrt{3 x}}{3-x}$.

So there you have it, guys! We've successfully simplified a radical expression by rationalizing the denominator. Remember, practice makes perfect, so keep honing your algebraic skills, and you'll be simplifying complex expressions like a pro in no time!

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