Simplifying Radicals: A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of radical expressions, specifically tackling the simplification of a complex one. We'll break down each step, making sure you not only understand the solution but also the underlying principles. Let's get started!

Understanding the Problem

Before we jump into the solution, let's clearly state the problem we're addressing. We need to simplify the following expression, given that $X$ is a natural number ($X \in N$):

$\sqrt[x]{\frac{\sqrt[x]{x^{x-1}}}{\sqrt[x]{\frac{x^{x+1}}{x^3}}}}$

This expression looks a bit intimidating at first glance, right? But don't worry, we'll take it step by step and make it much simpler. Our main goal here is to use the properties of exponents and radicals to reduce this expression to its simplest form. We'll be focusing on combining like terms, simplifying fractions within radicals, and ultimately, getting rid of as many radicals as possible. So, buckle up, and let's get into the nitty-gritty of simplifying this radical expression!

Step-by-Step Simplification

1. Simplify the Innermost Fraction

The first thing we're going to do, guys, is to attack the innermost fraction inside the radicals. We have $\frac{x^{x+1}}{x^3}$. Remember the rule of exponents that says when you divide powers with the same base, you subtract the exponents? So, we can rewrite this as:

$\frac{x^{x+1}}{x^3} = x^{(x+1) - 3} = x^{x-2}$

This simplifies our expression quite a bit. Now, instead of that messy fraction, we have a single term with an exponent. This is a crucial first step because it reduces the complexity inside the radicals, making the subsequent steps easier to manage. By applying this basic rule of exponents, we've already made significant progress toward simplifying the entire expression.

2. Substitute Back into the Expression

Now that we've simplified the innermost fraction, let's plug it back into our original expression. This is where we replace $\frac{x^{x+1}}{x^3}$ with $x^{x-2}$. Our expression now looks like this:

$\sqrt[x]{\frac{\sqrt[x]{x^{x-1}}}{\sqrt[x]{x^{x-2}}}}$

See how much cleaner that looks already? By substituting the simplified term, we're making the expression easier to work with. This step is all about keeping things organized and manageable. Remember, simplification is a process, and each step builds upon the previous one. So, by replacing the fraction with its simplified form, we're setting ourselves up for the next stage of simplification. Stay with me, guys, we're getting there!

3. Combine Radicals in the Denominator

Okay, guys, let's tackle those radicals in the denominator. We have $\sqrt[x]{x^{x-2}}$. We can rewrite this radical using fractional exponents. Remember, the $x$-th root of something is the same as raising it to the power of $\frac{1}{x}$. So, we can rewrite the denominator as:

$\sqrt[x]{x^{x-2}} = (x^{x-2})^{\frac{1}{x}}$

Now, when you raise a power to another power, you multiply the exponents. So, we have:

$(x^{x-2})^{\frac{1}{x}} = x^{\frac{x-2}{x}}$

This is a significant step! We've transformed the radical in the denominator into a simple exponent. This makes it much easier to combine terms and simplify further. By understanding the relationship between radicals and fractional exponents, we're able to manipulate the expression into a more manageable form. This is a key technique in simplifying radical expressions, so make sure you've got this one down!

4. Rewrite the Expression with Fractional Exponents

Let's rewrite the entire expression using fractional exponents. This will make it easier to manipulate and combine the terms. Remember, $\sqrt[x]{a} = a^{\frac{1}{x}}$. So, let's apply this to our expression:

$\sqrt[x]{\frac{\sqrt[x]{x^{x-1}}}{\sqrt[x]{x^{x-2}}}} = \left(\frac{(x^{x-1})^{\frac{1}{x}}}{x^{\frac{x-2}{x}}}\right)^{\frac{1}{x}}$

Now, let's simplify the numerator: $(x^{x-1})^{\frac{1}{x}} = x^{\frac{x-1}{x}}$. So, our expression becomes:

$\left(\frac{x^{\frac{x-1}{x}}}{x^{\frac{x-2}{x}}}\right)^{\frac{1}{x}}$

By converting the radicals to fractional exponents, we've opened the door to using the rules of exponents to further simplify the expression. This step is all about changing the representation of the expression to make it more amenable to simplification. We're setting the stage for the next step, where we'll combine the exponents and continue our journey to the final answer. You're doing great, guys! Keep going!

5. Simplify the Fraction Inside the Parentheses

Now, we're going to simplify the fraction inside the parentheses. Remember, when you divide powers with the same base, you subtract the exponents. So, we have:

$\frac{x^{\frac{x-1}{x}}}{x^{\frac{x-2}{x}}} = x^{\frac{x-1}{x} - \frac{x-2}{x}}$

Let's subtract the exponents:

$\frac{x-1}{x} - \frac{x-2}{x} = \frac{(x-1) - (x-2)}{x} = \frac{x - 1 - x + 2}{x} = \frac{1}{x}$

So, our fraction simplifies to:

$x^{\frac{1}{x}}$

This is a fantastic simplification! By applying the rules of exponents, we've reduced the complex fraction to a single term with a simple exponent. This step highlights the power of understanding and using exponent rules. We're getting closer and closer to the final simplified expression. Keep your eye on the prize, guys!

6. Apply the Outer Exponent

We're almost there, guys! Now, let's apply the outer exponent of $\frac{1}{x}$ to the simplified term inside the parentheses:

$\left(x^{\frac{1}{x}}\right)^{\frac{1}{x}}$

Remember, when you raise a power to another power, you multiply the exponents. So, we have:

$x^{\frac{1}{x} \cdot \frac{1}{x}} = x^{\frac{1}{x^2}}$

And there you have it! We've simplified the entire expression down to this. This final step is the culmination of all our hard work. By systematically applying the rules of exponents and radicals, we've transformed a complex expression into a simple, elegant form.

Final Answer

Therefore, the simplified expression is:

$x^{\frac{1}{x^2}}$

So, the correct answer is $1/x^2$.

Conclusion

Wow, we did it! We successfully simplified that complex radical expression. Remember, the key is to break down the problem into smaller, manageable steps. By understanding and applying the rules of exponents and radicals, you can tackle even the most intimidating expressions. Keep practicing, guys, and you'll become masters of simplification! This step-by-step approach is crucial not only for solving this particular problem but also for developing a problem-solving mindset that you can apply to other mathematical challenges. Keep up the great work, and remember, practice makes perfect! High-quality content like this helps solidify your understanding and builds your confidence in tackling complex mathematical problems. So, keep learning and keep simplifying!