1. **Problem Statement:** Solve the equation $$x^7 + x + 1 = 0$$ and find the value of $$5 \cdot x + \frac{1}{5 \cdot x}$$ for the root(s) of the equation. 2. **Understanding the problem:** The equation $$x^7 + x + 1 = 0$$ is a polynomial equation of degree 7. Finding exact roots analytically is generally very difficult for degree 5 or higher polynomials. 3. **Approach:** Since exact roots are hard to find, we consider the root $$x$$ that satisfies the equation and then evaluate the expression $$5x + \frac{1}{5x}$$. 4. **Key insight:** If we let $$y = 5x$$, then the expression becomes $$y + \frac{1}{y}$$. 5. **Note:** Without explicit roots, we cannot find a numeric value for $$5x + \frac{1}{5x}$$ exactly. However, if you want to approximate roots numerically, methods like Newton-Raphson or numerical solvers can be used. 6. **Summary:** The problem as stated requires numerical methods for root finding. Once a root $$x$$ is found, substitute into $$5x + \frac{1}{5x}$$ to get the value. **Final answer:** Cannot be expressed in closed form; numerical approximation needed.