CBSE Class 10 Answered

let us assume that root 3 is rational.

so we can find integers a and b (b not equal to zero) such that  root 3= a/b

suppose a and b have a common factor other than 1, then we can divide by the common factor and assume that a and b are coprime.

therefore, b* root3 = a

squaring both sides we get, 3 (b)^2 = (a)^2------------------------1

therefore (a)^2 is divisible by 3

By prime factorization we get that a is divisible by 3.

so we can write a=3c for some integer c

Substituting the value of a in above 1  we get,

3(b)^2= 9(c)^2

this implies (b)^2= 3(c)^2

therefore (b)^2 is divisible by 3 so b is divisible by 3.

so a and b have atleast 3 as a common factor.

But this contradicts the fact that a and b are coprime  which is a contradiction that root3 is rational.

Hence, root 3 is irrational.