To solve for \( x \) in the equation \( x^3 = 343 \), we take the cube root of both sides:
\[ x = \sqrt[3]{343} \]
Next, we can evaluate \( \sqrt[3]{343} \). Since \( 343 = 7 \times 7 \times 7 = 7^3 \), we have:
\[ \sqrt[3]{343} = 7 \]
Thus, the solution for \( x \) is:
\[ x = 7 \]