prove that (p)1/n is irrational when p is prime and n is greater than 1 - Maths - Real Numbers

Question

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

Ankush Jain · Accepted Answer

Let us consider that $p^{\frac{1}{n}}$ = x

Let us suppose that x is a rational number.

∴ x = $\frac{a}{b}$ where a and b are integers.

$Then, p^{\frac{1}{n}} = \frac{a}{b} \Rightarrow p = (\frac{a}{b})^{n} \Rightarrow p = \frac{a^{n}}{b^{n}}, which is a rational number.$

Since p is a prime then p is also an integer.

$∴ b^{n} = 1 or b^{n} = a^{m} where m < n$

Case 1:

If bⁿ = 1, then p = aⁿand p has factors other than p and 1 which contradicts our assumption that p is prime. So, b_n ≠ 1.

Case 2:

If bⁿ = a^m, where m < n, then p = a^{m + 1}..... aⁿand p still has factors other than p and 1 which again contradicts our assumption that p is prime.

So, our initial assumption that x is a rational number leads to contradiction in both the cases.

Hence, x is not a rational number i.e., it is a irrational number.

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

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