Question

Show that 3 + 2 √3 is an irrational number, where √3 is given to be irrational number. *​

Answer 1

Answer:

let us assume on the contrary that 3+2√3 is rational. then there exists co prime positive integers a and b such that

3+2√3 = a/b

=> 2√3 = (a/b) - 3

=> √3 = (a - 3b)/2b

=> √3 is rational [ as a, b are integers, so,(a - 3b)/2b is a rational no.]

this contradicts the fact that √3 is irrational. so, our assumption is incorrect.

hence, 3+2√3 is an irrational number.