Question

Prove that √p + √q is irrational, where p, q are
q are primes.​

Answer 1

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lets assume that √p is rational,

⇒ √p = a/b ( where 'a' and 'b' are co primes, meaning they don't have any common factors except for 1)

From squaring both sides,

p = a²/b²

⇒pb² = a²

⇒ b² = a²/p

Since 'p' divides a², it also divides 'a' meaning 'a' has a factor of p

Let 'a' = pm (where m is a positive integer) ⇒ a² = p²m²

Now, pb² = a²

pb² = p²m²

pb²/p²= m²

b²/p =m²

∴ 'p' divides 'b' ⇒ 'b' also has a factor 'p'

∴ 'a' and 'b' are not co primes and our assumption was wrong

⇒ √p is irrational

Similarly √q is irrational

∴⇒ √p + √q is irrational