The given equation is:

$x^3 = \frac{8}{729}$

To solve for $x$, we take the cube root of both sides:

$x = \sqrt[3]{\frac{8}{729}}$

We can simplify this by separately taking the cube roots of the numerator and the denominator:

[ x = \frac{\sqrt[3]{8}}{\sqrt[3]{729}} = \frac{2}{9} ]

Thus, the solution is:

$x = \frac{2}{9}$

Would you like further clarification on any step?

Here are 5 related questions to deepen your understanding:

1. What is the general method for solving cubic equations like this one?
2. How do cube roots relate to powers of numbers?
3. What happens if the cube on the left side was negative, like $x^3 = -\frac{8}{729}$?
4. How does solving fractional exponents differ from regular integers?
5. Could this problem have been solved graphically, and if so, how?

Tip: For fractional powers, it’s often easier to split the problem into separate parts (like handling the numerator and denominator separately, as we did here).