Closed form solution for constant exponent in sum I am trying to solve for $\alpha$ in the following equation:
$$ 0.80 = \frac{1}{3} \left( X_1^\alpha + X_2^\alpha + X_3^\alpha \right)$$
Right now I just use Excel and solver to find a numerical solution to problems like these.  Seems like there should be a way to obtain a closed form solution.
This is an exponential curving example for grades that sets the average post-curved grade to 80%. Let me make it a bit more general:
$$ 0.80 = \frac{1}{n} \sum_i^n X_i^\alpha$$ where $n$ is the number of students that took the exam, $0 < X_i < 1$ is an exam grade for student $I$, and $0 < \alpha \le 1$ is a scaling factor.  Everything is known except $\alpha$.
This type of curving "gives to the needy and not the greedy" in that lower scoring students receive more of a bump yet the rank ordering is unchanged.

There is no closed form, for a provable reason.
Setting $t=X_1^\alpha$, the equation can be rewritten
$$t+t^p+t^q=c$$ where $p:=\dfrac{\log X_2}{\log X_1},q:=\dfrac{\log X_3}{\log X_1}$.
In general these exponents are fractional, but they can as well be integer. And it was proven that there is no general formula for the roots of a polynomial.
