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I am assuming that, n is a prime number, because you can express composite numbers to prime by enlarging the exponent.
a = nn^x-1 = n
I don't know how to prove or disprove it, but if x is odd then,
a is always multiplied by a prime number, which square rooted are irrational.
For example n=5; x=11
√5^11 = √5^105 = 5^5√5
The prime number is left in the sqrt. Is my hint correct? I don't think it completely proves this theory.

If x is even, then halving the exponent is good enough.

N and x values are acceptable as disproving.
Thanks.

Thomas Andrews
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    Please use MathJax when formatting your question: https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – Gary Oct 31 '21 at 23:09
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    How about $4^3$? Statement is true if $n$ is a square free number. – herb steinberg Nov 01 '21 at 00:41

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