A Nice Math Olympiad exponential question| Solve for x 

Excellent presentation my dear

X^6=2^6 X^6=64 X=±2 X=±1±Sqrt[3]i It’s in my head.

x = -2 also a solution. Haven't we missed some solutions here (including complex ones)? A good idea to check for mistakes / missed assumptions, when you don't get an obvious answer...

Greetings. Happy Sunday night. Thanks always for sharing. I instantaneously realize that X^6=2^6 is X^6-2^6=0, (X^3)^2-(2^3)^2=0, (X^3-2^3)(X^3+2^3)=0, from which, X^3-2^3= (X--2)(X^2 +2X+4)=0, and from X^3+2^3, we have (X+2)(X^2-2X+4)=0. So far when X-2=0, X=2, and when X+2=0, X=-2 for two of the roots of the expression. The other four roots are complex roots and are disregarded for presentation. However, I have calculated the results. Blessings.

(±2)^6=64 (±1±Sqrt[3]i)^6=64 64=64 solution

JJ , your lesson is very sweet. Thank you very much my dear.

(x^2)^3-(2^2)^3 =?