To rationalize the numerator of the expression:

$\frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

we should multiply both the numerator and the denominator by the conjugate of the numerator, which is $\sqrt{x} + 1$. Here are the steps:

Step 1: Multiply by the Conjugate

Multiply both the numerator and denominator by $\sqrt{x} + 1$:

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

Step 2: Simplify the Numerator

Expand the numerator $(\sqrt{x} + 1)^2$:

$(\sqrt{x} + 1)^2 = (\sqrt{x})^2 + 2 \cdot \sqrt{x} \cdot 1 + 1^2 = x + 2\sqrt{x} + 1$

So, the numerator becomes $x + 2\sqrt{x} + 1$.

Step 3: Simplify the Denominator

Use the difference of squares formula for the denominator $(\sqrt{x} - 1)(\sqrt{x} + 1)$:

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

Final Expression

Now we have:

$\frac{x + 2\sqrt{x} + 1}{x - 1}$

This is the rationalized form of the original expression.

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

Here are some related questions you might explore:

1. How would you rationalize the denominator instead of the numerator?
2. Can you apply a similar process to rationalize expressions with cube roots?
3. What is the benefit of rationalizing an expression in mathematics?
4. How does the difference of squares formula help in rationalizing?
5. Are there other methods to simplify expressions like this one?

Tip: When rationalizing, always look for the conjugate, as it eliminates square roots when you multiply.