Prove that √3+√5 irrational

Let us suppose that √3+√5 is rational. Let √3+√5=a is rational. Therefore, √3=a−√5 On squaring both sides, we get, (√3)2=(a−√5)2 ⇒ 3=a2+5−2a√5 [∵ (a−b)2=a2+b2–2ab] ⇒ 2a√5=a2+2 Therefore, √5=a2+22a As the right hand side is rational number while √5 is irrational. It contradicts with our assumption. Hence, √3+√5 is irrational. |