Numerical error

Question

My brother told me that when he did this operation in the calculator: "the 2^33 root of one natural number", he obtains exactly 1. He asked me why (because I am a math student) but I don't know the answer. Do you?

Answer 1

According to Bernoulli's inequality, for $r\in(0,1]$ and $x>0$, $1\le(1+x)^r\le1+rx$.

Therefore, letting $n=1+x$, $1\le n^r\le 1+r(n-1)$.

With $r=2^{-33}$ and $n$ a number that is in the range to be fully displayed on the calculator ,

this forces $n^r$ to be so close to $1$ that, as saulspatz commented, it displays as $1$.