To solve the equation
\[ \frac{x^{10}}{3} = 27 x^{\frac{1}{3}}, \]
we can first eliminate the fraction by multiplying both sides by 3:
\[ x^{10} = 81 x^{\frac{1}{3}}. \]
Next, we can rearrange this equation to isolate one side:
\[ x^{10} - 81 x^{\frac{1}{3}} = 0. \]
Now, we can factor out \( x^{\frac{1}{3}} \):
\[ x^{\frac{1}{3}}(x^{\frac{29}{3}} - 81) = 0. \]
Setting each factor to zero gives us two cases to solve:
- \( x^{\frac{1}{3}} = 0 \)
- \( x^{\frac{29}{3}} - 81 = 0 \)
From the first case, \( x^{\frac{1}{3}} = 0 \) implies:
\[ x = 0. \]
From the second case:
\[ x^{\frac{29}{3}} = 81. \]
We can rewrite 81 as \( 3^4 \):
\[ x^{\frac{29}{3}} = 3^4. \]
Taking both sides to the power of \( \frac{3}{29} \):
\[ x = (3^4)^{\frac{3}{29}} = 3^{\frac{12}{29}}. \]
Thus, the complete solution for \( x \) is:
\[ x = 0 \quad \text{or} \quad x = 3^{\frac{12}{29}}. \]
To summarize, the solutions are:
\[ \boxed{0} \quad \text{and} \quad \boxed{3^{\frac{12}{29}}}. \]