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Proving that (p)1/n is Irrational when p is Prime and n>1
xn - p = 0
or, equivalently, x = (p)1/n.
and specify that n > 1.11 Prove that x is an irrational number.
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 12
Special case: If m = n - 1, then p = a, and bn = pn - 1, or p = b(n/[n - 1]) = b. But if this latter statement is true, then p = 1, and we violate the assumption that p is any prime.
Since the assumption that x is a rational number leads to contradictions in all possible cases, we must conclude that x is irrational.
11 Notice that, for n = 1, x = p, and p = p/1 = p/ = /, which is a rational number. The proof for irrationality is only valid when n > 1.
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