Question 679089
recall that, by definition, an exponent can be any number, real or complex, negative or positive, rational or {{{irrational}}}, algebraic or transcendental.

{{{x^(pi)}}} is an example of an irrational exponent

or {{{2^sqrt(2)}}}....how to do this one?

we will do it by {{{approximating}}} the {{{irrational}}} exponent by {{{rational}}} exponents that are closer and closer to the irrational one

In example {{{2^sqrt(2)}}}, you want the exponent {{{sqrt(2)}}} to be {{{approximately}}} equal to {{{ 1}}}, so the first approximation to {{{2^sqrt(2)}}} is {{{2^1 = 2}}}.  

or, go one step further with {{{sqrt(2)=1.4 =14/10 =7/5}}}, so the second approximation to {{{2^sqrt(2)}}} is {{{2^(7/5)=root(5,2^7)}}} which is approximately {{{2.639}}}  

or three significant figures, 
{{{sqrt(2) = 1.41 = 141/100}}}, so you have to take a {{{root(100,2^141)}}} which is approximately {{{2.65737}}}  Continuing this way, we get the sequence of approximations:

{{{2}}}, {{{2.639}}}, {{{2.6574}}}, {{{2.66475}}}, {{{2.665119}}}, {{{2.6651375}}}, {{{2.66514310}}}, ...