Prove that the following are irrational.

3 + 2√5

Let us assume on the contrary that 3 + 2√5 is rational. Then, there exist co-prime positive integers a and b such that

3 + 2√5

⇒

⇒

⇒ is rational [∵ a,b are integers ∴ is a rational]

This contradicts the fact that is irrational. So, our supposition is incorrect.

Hence, is an irrational number.