1. **State the problem:** Simplify the expression $y = x \sqrt{\frac{x^2+1}{x^2-1}}$. 2. **Recall the formula and rules:** The square root of a fraction is the fraction of the square roots: $$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$ provided $b \neq 0$. 3. **Rewrite the expression:** $$y = x \cdot \frac{\sqrt{x^2+1}}{\sqrt{x^2-1}}$$ 4. **Analyze the domain:** - The denominator $\sqrt{x^2-1}$ requires $x^2 - 1 > 0 \Rightarrow |x| > 1$. 5. **Simplify if possible:** No further simplification is possible because $x^2+1$ and $x^2-1$ do not factor nicely to simplify the square roots. 6. **Final simplified form:** $$y = \frac{x \sqrt{x^2+1}}{\sqrt{x^2-1}}$$ This is the simplified expression with domain $|x| > 1$ to keep the expression real.