To simplify the expression $X = \frac{1}{\sqrt{5} + 2}$, we can rationalize the denominator. Here's how it's done:

Step 1: Multiply the numerator and the denominator by the conjugate of the denominator.

The conjugate of $\sqrt{5} + 2$ is $\sqrt{5} - 2$.

$X = \frac{1}{\sqrt{5} + 2} \times \frac{\sqrt{5} - 2}{\sqrt{5} - 2}$

Step 2: Simplify the expression.

$X = \frac{\sqrt{5} - 2}{(\sqrt{5} + 2)(\sqrt{5} - 2)}$

Step 3: Use the difference of squares in the denominator.

The denominator becomes: $(\sqrt{5})^2 - 2^2 = 5 - 4 = 1$

So the expression simplifies to: $X = \sqrt{5} - 2$

Final Answer:

$X = \sqrt{5} - 2$

Would you like any details or have further questions on this solution?

Related Questions:

1. How do you find the conjugate of a binomial?
2. What is the difference of squares, and how is it used in rationalizing denominators?
3. How can you rationalize denominators with more complex expressions?
4. What are other methods to simplify irrational expressions?
5. Can you provide examples of rationalizing different types of denominators?

Tip:

Always remember that rationalizing the denominator is a helpful technique for simplifying expressions involving roots. It often makes the expression easier to work with in further calculations.