You can pretty much do this one in your head:
My answer = in the comments.
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So, we have
ReplyDeletex^x - x^(1/x) = 0
Which means
x^x = x^(1/x)
Since the bases are the same, we can discount them, so
x = 1/x
Multiply by x on both sides, and
x^2 = 1
So x = ±1.
QED.