I recently stumbled onto one old math olympiad problem which was rather intriguing. We need to solve for value of $x$ in following equation:

$$ x^{x^4} = 64

$$

Math is beautiful and its beauty lies in eyes of beholder. A simple looking equation when we go on solving it requires lot of brainstorming. Let’s try to solve this equation.

Math solution:

We are given following equation:

$$ x^{x^4} = 64 $$

Taking power of 4 on both sides:

$$ (x^{x^4})^4 = 64^4 $$

which is equivalent of:

$$ x^{x^4*4} = (8^2)^4 $$

on further simplification:

$$ ({x^4})^{x^4} = 8^8 $$

which gives us:

$$ x^4 = 8 $$

So our answer is $x = 2^{3/4}$ = 1.681793

Elegant solution! But not intuitive and the same heuristic may not work for similar other equations. So, how to tackle such kind of problems.