find the number of solutions

One way to solve this problem is to take logs of both sides of the original equation to obtain $x = lnx^4$ . Now if one differentiates both sides with respect to x we now get $1 = 4/x$ , which the logarithmic curve has a tangent line that's parallel to $y = x$ at $x = 4$ . This tangent line is expressible as $y - ln256 = x - 4$ , or $y = x + (ln256 - 4)$ which runs parallel and above $y = x$ .

Since the point $(x, y ) = (4, ln 256)$ lies above $(4, 4)$ , there exists two points $(a, a)$ and $(b, b)$ on the line $y = x$ which are intersection points with $y = ln x^4$ (by the Mean-Value Theorem). Since the logarithmic curve grows at a more slower rate than the linear function (i.e. $1 > 4 /x$ for $x = [b, \infty)$ , there are no more intersection points to be found.

Thus, $n = 2$ and $e^2 + e = 10.10$ .