If a number has a finite but nonzero fractional part, so do all its powers. I recently ran across a proof in [1] that is shorter than I expected.
Theorem: Suppose r is a real number that is not an integer, and the decimal part of r terminates. Then rk is not an integer for any positive integer k.
Proof: The number r can be written as a reduced fraction a / 10m for some positive m. If s = rk were an integer, then
10mk s = ak.
Now the left side of this equation is divisible by 10 but the right side is not, and so s = rk must not be an integer. QED.
The only thing special about base 10 is that we most easily think in terms of base 10, but you could replace 10 with any other base.
What about repeating decimals, like 1/7 = 0.142857142857…? They’re only repeating decimals in our chosen base. Pick the right base, i.e. 7 in this case, and they terminate. So the theorem above extends to repeating decimals.
[1] Eli Leher. √2 is Not 1.41421356237 or Anything of the Sort. The American Mathematical Monthly, Vol. 125, No. 4 (APRIL 2018), page 346.