Spivak Calculus Chapter 2,Number 18, Connection with Number 17 In Spivak's Calculus, Problem 17 has you $(1)$ verify that all natural numbers are factorable into a product of purely prime numbers and $(2)$ has you show that $n^{\frac{1}{k}}$ is irrational unless $n=m^k$ for some natural number $m$. 
Problem 18 asks 
Prove that if $x$ satisfies
$$x^n+a_{n-1}x^{n-1}+...+a_0 = 0$$
then $x$ is irrational unless it is an integer. How is this a generalization of Problem 17? 
I know the solution to Problem 18, but I do not see how it is a generalization of Problem 17. I feel that the main issue is that I do not understand if it has to do primarily with the factorization into primes, the irrationality of $k_{th}$ roots, or relies on both. 
I would very much appreciate a hint to get started on seeing the connection. 

If $k,n\in\mathbb N$, consider the polynomial $x^k-n$. Problem 17 is equivalent to the assertion that if this polynomial has a rational root, then it is actually an integer one.
