Can index in a radix be $1$?

Question

I am stuck at a very basic question

I had a true/false question where the statement was

"if $n$ is a natural number and $x$ a prime number $$\sqrt[n]{x}$$ is always irrational "

I was of the impression that the answer would be yes, but what if $n = 1$. So basically my question is can $n$ take value as one ? And if yes what would be result in that case ?

Answer 1

The statement

if $n$ is a natural number and $x$ a prime number, $\sqrt[n] x$ is always irrational

Is actually false since if $n=1$ then $$\sqrt[1] x=x^{1/1}=x$$

Of course if the statement was

if $n$ is a natural number larger than $1$ and $x$ a prime number, $\sqrt[n] x$ is always irrational

Then it would be true

---

In general

$$\sqrt[n] x=x^{1/n}$$

This means that $n$ can never be $0$ because that would result in division by zero.
There are also some other cases where the exponent is undefined (unless you're working with complex numbers)

When $x$ is positive, $x^n$ is always defined.

When $x$ is zero, $x^n$ is defined to be $0$ unless $n\le0$

When $x$ is negative, $x^n$ is only defined if $n$ is an integer or $n$ is fraction with an odd denominator.

Answer 2

The radical is only a short-cut notation for fractional powers. Commonly, you write $\sqrt[n]{x}$ to talk about $x^{\frac{1}{n}}$. Do you know what are fractional powers? Take a look at Wiki Page about Powers.