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Take log on both sides.
(x^(-x))log [( a^x)^x] = [(1/8)^(1/2)]log a,
(x^2)(x^(-x))log a = [(1/2)^(3/2)]log a,
log a get cancelled
x^(2-x) = [(1/2)^(3/2)]
x^(2-x) = [(1/2)^((4 - 1)/2)]
x^(2-x) = [(1/2)^(2 - 1/2)]
Comparing x = (1/2)