Prove underroot 3+underroot 5 is irrational

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unlock 35 lakh questions What's your doubt? 10th >Maths >Real Numbers >Revisiting Irrational Numbers >Prove that √(3) + √(5) is... Maths Bookmark Prove that 3 ​ + 5 ​ is irrational. Easy share Share Answer To prove : 3 ​ + 5 ​ is irrational. Let us assume it to be a rational number. Rational numbers are the ones that can be expressed in q p ​ form where p,q are integers and q isn't equal to zero. 3 ​ + 5 ​ = q p ​ 3 ​ = q p ​ − 5 ​ squaring on both sides, 3= q 2 p 2 ​ −2. 5 ​ ( q p ​ )+5 ⇒ q (2 5 ​ p) ​ =5−3+( q 2 p 2 ​ ) ⇒ q (2 5 ​ p) ​ = q 2 2q 2 −p 2 ​ ⇒ 5 ​ = q 2 2q 2 −p 2 ​ . 2p q ​ ⇒ 5 ​ = 2pq (2q 2 −p 2 ) ​ As p and q are integers RHS is also rational. As RHS is rational LHS is also rational i.e 5 ​ is rational. But this contradicts the fact that 5 ​ is irrational. This contradiction arose because of our false assumption. so, 3 ​ + 5 ​ irrational.