1. **Stating the problem:** Simplify the expression $$(\sqrt{x} - 1)(\sqrt{x+1} + \sqrt{x-1}) - \sqrt[3]{x\sqrt{x}}$$ 2. **Recall formulas and rules:** - Difference of squares: $ (a - b)(a + b) = a^2 - b^2 $ - Simplify radicals by expressing powers as fractional exponents. 3. **Simplify the first product:** $$(\sqrt{x} - 1)(\sqrt{x+1} + \sqrt{x-1})$$ We cannot directly apply difference of squares here because the terms inside the second parentheses are different. 4. **Rewrite the expression:** Let $A = \sqrt{x}$, then the expression is $$(A - 1)(\sqrt{x+1} + \sqrt{x-1}) - \sqrt[3]{x A}$$ 5. **Expand the first product:** $$A \sqrt{x+1} + A \sqrt{x-1} - \sqrt{x+1} - \sqrt{x-1}$$ 6. **Rewrite the cube root term:** $$\sqrt[3]{x \sqrt{x}} = \sqrt[3]{x \cdot x^{1/2}} = \sqrt[3]{x^{3/2}} = x^{\frac{3/2}{3}} = x^{1/2} = \sqrt{x} = A$$ 7. **Substitute back:** The expression becomes $$A \sqrt{x+1} + A \sqrt{x-1} - \sqrt{x+1} - \sqrt{x-1} - A$$ 8. **Group terms:** $$A(\sqrt{x+1} + \sqrt{x-1} - 1) - (\sqrt{x+1} + \sqrt{x-1})$$ 9. **Final simplified form:** $$\boxed{\sqrt{x}(\sqrt{x+1} + \sqrt{x-1} - 1) - (\sqrt{x+1} + \sqrt{x-1})}$$ This is the simplified expression.