To solve the rational equation given, we first need to understand it clearly. It seems that the equation is structured as follows:
\[ \frac{3x + x - 3x}{3} = \frac{1}{x^2} \]
Let's simplify the left-hand side:
- Combine the terms in the numerator: \[ 3x + x - 3x = x \]
Thus, we rewrite the left side: \[ \frac{x}{3} = \frac{1}{x^2} \]
Next, let's cross-multiply to eliminate the fractions: \[ x \cdot x^2 = 3 \cdot 1 \] This simplifies to: \[ x^3 = 3 \]
Now, we take the cube root of both sides: \[ x = \sqrt[3]{3} \]
Thus, the solution to the equation is: \[ x = 3^{1/3} \]
It's always good to check for extraneous solutions, especially when dealing with rational equations. In this case, \(x = \sqrt[3]{3}\) is defined and does not make any denominators zero in the original equation.
Therefore, the final solution is: \[ \boxed{\sqrt[3]{3}} \]