Question

$prove that ( \sqrt{2n + 1 + } \sqrt{2n + 3)} is irrational when p is prime and n > 1$
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Answer 1

   

Let us assume to reach our contradiction that (p)^(1/n) is rational. So that (p)^(1/n) can be written in the form of a/b, where a, b are co-prime integers and b > 0. Such that,

⇒ (p)^(1/n) = a/b

⇒ ((p)^(1/n))ⁿ = (a/b)ⁿ

⇒ p = aⁿ/bⁿ

⇒ pbⁿ = aⁿ

Here seems that aⁿ is a multiple of p. As a is an integer and n > 1, if aⁿ is a multiple of p, then so will be aⁿ. So let a = px. Such that,

⇒ pbⁿ = aⁿ

⇒ pbⁿ = (px)ⁿ

⇒ pbⁿ = pⁿ xⁿ

⇒ bⁿ = pⁿ⁻¹ xⁿ

As n > 1, n - 1 > 0, so that pⁿ⁻¹ is an integer. Here seems that bⁿ is a multiple of pⁿ⁻¹, which is a multiple of p. So bⁿ is also a multiple of p. As b is also integer, if bⁿ is multiple of p, then so will be b.

But this contradicts our earlier assumption that a, b are co-prime integers, because here seems that both a and b are divisible by p.

So our assumption is contradicted.

∴  (p)^(1/n) is irrational.

Hence proved!!!

Thank you...