Question

Prove that \(3 + 2\sqrt 5\) is irrational.

Answer 1

Let \(3+2\sqrt 5\) be rational.
Therefore, we can find two co-prime integers \(a, b\  (b ≠ 0)\) such that
\(3+2\sqrt 5=\dfrac{a}{b}\)

\(⇒2\sqrt 5=\dfrac{𝑎}{𝑏}−3\) 

\(⇒\)\(\sqrt 5=\dfrac{1}{2}(\dfrac{𝑎}{𝑏}−3)\)

Since a and b are integers, \(\dfrac{1}{2} (\dfrac{a}{b} −3)\) will also be rational, and therefore, \(\sqrt 5\) is rational.
This contradicts the fact that \(\sqrt 5\) is irrational. Hence, our assumption that \(3+2\sqrt 5\)  is rational is false. Therefore, \(3+2\sqrt 5\) is irrational.