Question

prove that (p)1/n is irrational when p is prime and n is greater than 1.

Answer 1

Proving that (p)1/n is Irrational when p is Prime and n>1 

Problem:
Let p be any prime number, and let p satisfy the equation

xn - p = 0

or, equivalently, x = (p)1/n.

and specify that n > 1.11 Prove that x is an irrational number.

 

Solution:
The proof that, under this condition, x is irrational will be done indirectly by assuming that x is rational, then showing that this assumption leads to a contradiction.

Let x be rational; i.e., let x = a/b where a and b are integers. Then:

p = xn = an/bn = a rational number.

Since p is prime, then p is an integer. Thus, either:

bn = 1 or bn = am where m < n 12If bn = 1, then p = an, and p has factors other than p and 1,13 violating the assumption that p is prime. Therefore, bn 1.If bn = am, where m < n, then p = am+1… an, and p still has factors other than p and 1,14 violating the assumption that p is prime.