The given mathematical expression appears to be:

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

Let's simplify it step by step:

Step 1: Convert Division into Multiplication

Rewriting the division as multiplication by the reciprocal:

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

Step 2: Multiply Fractions

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

Step 3: Express in a Common Denominator

Rewriting $1$ as $\frac{x-1}{x-1}$:

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

Step 4: Combine the Fractions

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

This is the simplified form:

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

Would you like further clarification or examples? Here are some related questions:

1. How do you simplify complex fractions?
2. What are the rules of fraction division?
3. How do you rationalize expressions with square roots?
4. What happens when $x = 1$ in this expression?
5. How can this expression be applied in calculus?

Tip: Always rewrite division as multiplication by the reciprocal to simplify expressions efficiently.