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The given expression is:

\[
\frac{x}{\sqrt{x^2 - y^2}} - \frac{\sqrt{x^2 - y^2}}{x}
\]

To rationalize and simplify this expression, we'll proceed step by step.

### Step 1: Combine the terms into a single fraction.
We want to express both terms with a common denominator:

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

### Step 2: Simplify the expression.
Now combine both fractions under the common denominator:

\[
\frac{x^2 - (x^2 - y^2)}{x\sqrt{x^2 - y^2}} = \frac{x^2 - x^2 + y^2}{x\sqrt{x^2 - y^2}} = \frac{y^2}{x\sqrt{x^2 - y^2}}
\]

### Final Result:
The simplified expression is:

\[
\frac{y^2}{x\sqrt{x^2 - y^2}}
\]

Would you like further details or any specific part explained?

Here are 5 related questions to explore further:
1. How do you handle square roots in rationalizing complex fractions?
2. What is the significance of rationalizing denominators?
3. How would you simplify an expression with higher powers inside a square root?
4. What are common strategies for finding a common denominator in algebraic fractions?
5. How can trigonometric identities be applied to similar types of expressions?

**Tip:** Always check for common factors or terms that can cancel out when simplifying algebraic fractions!

Rationalize and simplify the expression: \[ \frac{x}{\sqrt{x^2 - y^2}} - \frac{\sqrt{x^2 - y^2}}{x} \]

Learn how to rationalize and simplify the expression \frac{x}{\sqrt{x^2 - y^2}} - \frac{\sqrt{x^2 - y^2}}{x} with step-by-step instructions. Explore core concepts such as radicals, simplifying expressions, and rationalizing denominators. Suitable for high school students in Grades 10-12.

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