# Ex 1.3, 1 - Class 10

Last updated at April 20, 2018 by Teachoo

Last updated at April 20, 2018 by Teachoo

Transcript

Ex 1.3 , 1 Prove that β5 is irrational. We have to prove β5 is irrational Let us assume the opposite, i.e., β5 is rational Hence, β5 can be written in the form π/π where a and b (bβ 0) are co-prime (no common factor other than 1) Hence, β5 = π/π β5b = a Squaring both sides (β5b)2 = a2 5b2 = a2 π^2/5 = b2 Hence, 5 divides a2 So, 5 shall divide a also Hence, we can say π/5 = c where c is some integer So, a = 5c Now we know that 5b2 = a2 Putting a = 5c 5b2 = (5c)2 5b2 = 25c2 5b2 = 25c2 b2 = 1/5 Γ 25c2 b2 = 5c2 π^2/5 = c2 Hence 5 divides b2 So, 5 divides b also By (1) and (2) 5 divides both a & b Hence 5 is a factor of a and b So, a & b have a factor 5 Therefore, a & b are not co-prime.Hence, our assumption is wrong β΄ By contradiction, β5 is irrational

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.