數學高手請進...(10分)

For such question, we have to solve it by means of Newton's-Raphson method which is a method of approximation of root as follows:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Maths/Newtonmethod.jpg

From the graph above, we can see that the general expression of each iterative value of root is:
xn+1 = xn - [F(xn)/F'(xn)]
Usually the iteration can top if |xn+1 - xn| < 10-4.
For the estimation of x0, find it such that F(x0) is reasonably small.
Now, let F(x) = xx - 16777216, then
F'(x) = xx(ln x + 1) by means of logarithmic differentiation.
So for solving F(x) = 0, we start with x0 = 8.5 that F(x0) = 64666741
So using the above theory, we can perform the iteration as follows:
x1 = 8.2488
x2 = 8.0761
x3 = 8.0084
x4 = 8.0001
x5 = 8.0000
x6 = 8.0000
Therefore, we can see that the root can be approximated to x = 8.0000 (In fact this can be solved by direct substitution).
For any equation of the form xx - K = 0, where K is a positive constant, this method can be used to find out the approx. value of x.

對於 x^x=y for constant y 的問題，
係無分析方法解決的，

要用的係 Newton's method 之類的約數推算方法。
如果你未學過微積分，看 weblink 也不可能明白，這樣我也無法子教你這是什麼，因為數學基礎未夠。