1. **Stating the problem:** Solve the equation $$\sqrt[10]{\frac{x^2 + \frac{x}{y}}{}} = \sqrt[10]{\left(x + \frac{1}{x^2}\right)^5} \cdot \sqrt{x}$$ given in exercise 123, page 690. 2. **Understanding the equation:** The equation involves 10th roots and square roots. We want to simplify and solve for $x$ (assuming $y$ is a parameter or constant). 3. **Rewrite the equation clearly:** $$\sqrt[10]{x^2 + \frac{x}{y}} = \sqrt[10]{\left(x + \frac{1}{x^2}\right)^5} \cdot \sqrt{x}$$ 4. **Express the right side as a single root:** Recall that $$\sqrt[10]{a^5} = a^{\frac{5}{10}} = a^{\frac{1}{2}} = \sqrt{a}$$. So, $$\sqrt[10]{\left(x + \frac{1}{x^2}\right)^5} = \sqrt{x + \frac{1}{x^2}}$$. Therefore, the right side becomes: $$\sqrt{x + \frac{1}{x^2}} \cdot \sqrt{x} = \sqrt{x \left(x + \frac{1}{x^2}\right)}$$ 5. **Simplify inside the square root:** $$x \left(x + \frac{1}{x^2}\right) = x^2 + \frac{x}{x^2} = x^2 + \frac{1}{x}$$ 6. **Rewrite the equation:** $$\sqrt[10]{x^2 + \frac{x}{y}} = \sqrt{x^2 + \frac{1}{x}}$$ 7. **Raise both sides to the 10th power to eliminate the 10th root:** $$x^2 + \frac{x}{y} = \left(\sqrt{x^2 + \frac{1}{x}}\right)^{10}$$ 8. **Simplify the right side:** Since $$\left(\sqrt{A}\right)^{10} = A^5$$, we get $$x^2 + \frac{x}{y} = \left(x^2 + \frac{1}{x}\right)^5$$ 9. **Final equation to solve:** $$x^2 + \frac{x}{y} = \left(x^2 + \frac{1}{x}\right)^5$$ 10. **Interpretation:** This is a complex equation involving $x$ and $y$. Without additional information about $y$, the equation cannot be solved explicitly for $x$ in simple closed form. **Summary:** The original equation simplifies to $$x^2 + \frac{x}{y} = \left(x^2 + \frac{1}{x}\right)^5$$ which is the key relationship to analyze or solve further given values or constraints on $y$. **Answer:** $$\boxed{x^2 + \frac{x}{y} = \left(x^2 + \frac{1}{x}\right)^5}$$