How many roots (include complex roots) are there in x^√2 = 1

z^√2=1
z=cis(2kπ/√2)=cis(√2 kπ) where k is integer

since √2 is irrational, √2 kπ will never repeat,
therefore there is infinitely many roots.

suppose x be the complex number,the index of the complex number x indicate the number of roots,but we cannot say that there are √2 roots,the number of roots should be an integer!Therefore,there is no complex root.
The question should be solved by the simple method:
x√2 = 1
(√2)logx = log1
(√2)logx = 0
logx = 0
x=1
The only root is 1