Question
$$x^3-\frac{5}{15} = 8$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} x^3-\frac{5}{15} &= 8&& \text{multiply ALL terms by } \color{blue}{ 15 }. \\[1 em]15x^3-15\cdot\frac{5}{15} &= 15\cdot8&& \text{cancel out the denominators} \\[1 em]15x^3-5 &= 120&& \text{move all terms to the left hand side } \\[1 em]15x^3-5-120 &= 0&& \text{simplify left side} \\[1 em]15x^3-125 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using qubic formulas.
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Equations Solver