The given equation is:

[ \sqrt[3]{x - 10} + \sqrt[3]{x - 17} = 3 ]

Solution:

1. Let substitution simplify the problem:
   Let $a = \sqrt[3]{x - 10}$ and $b = \sqrt[3]{x - 17}$.
   From the given equation:
   $a + b = 3$

2. Relating the terms $a^3$ and $b^3$:
   Since $a^3 = x - 10$ and $b^3 = x - 17$, subtracting gives:
   $a^3 - b^3 = 7$
   Using the identity $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, we can rewrite this as:
   $(a - b)((a + b)^2 - 3ab) = 7$
   Substituting $a + b = 3$, we get:
   $(a - b)(9 - 3ab) = 7$

3. Solving for $a$ and $b$:
   Let $a - b = t$. Then, from $a + b = 3$ and $a - b = t$, we can express $a$ and $b$ as:
   $a = \frac{3 + t}{2}, \quad b = \frac{3 - t}{2}.$

   Substitute $a = \frac{3 + t}{2}$ and $b = \frac{3 - t}{2}$ into the equations above to