All Natural Solution

Number Theory Level 2

For some natural numbers $k$ , $p$ & $x$ ,

Can $x^{\frac{1}{k}}$ ever be rational if $x$ cannot be written as $p^{k}$ ?

Note: the natural numbers set does not include 0.

No Yes

1 solution

Joseph Wilson
Aug 16, 2018

Suppose there is a rational number $S = \frac{a}{b}\ = x^{\frac{1}{k}}$ , where $a$ and $b$ are coprime natural numbers:

$\frac{a^{k}}{b^{k}} = x$

As $a$ and $b$ do not have any proper factors in common, the denominator, $b$ will never be able to cancel with factors in the numerator.

Therefore $x$ must always be written as an irreducible fraction, which is not a natural number, so the supposition that $S$ exists is wrong.