AIME 1986 · 第 1 题

Solution

Let $y = \sqrt[4]{x}$. Then we have $y(7 - y) = 12$, or, by simplifying, **

$$y^2 - 7y + 12 = (y - 3)(y - 4) = 0.$$ **

This means that $\sqrt[4]{x} = y = 3$ or $4$.

Thus the sum of the possible solutions for $x$ is $4^4 + 3^4 = \boxed{337}$.