Question

prove root3 is irrational and hence 3/2root3 is irrational

Answer 1

To prove root 3 is irrational ,let us assume that root 3is rational .
rational number = a/b
√3=a/b
by squaring on both sides
(√3)²= (a/b)²
3=a²/b²
3b²=a²_1
if a² is divisible by a.
then a is divisible by a.
then 3b² is divisible by a.(if p² is divisible by a then p is divisible by a )
let a=3c
squaring on both sides
a²=(3c)²
a²=9c²_2
if a² is divisible by a.
then a is divisible by a.
then 9c² is divisible by a.
by 1 and 2
we get a²=3b²=a²=9c²
cancel a² (common)
3b²=9c²
b²=3c²
if b² is divisible by b.
then b is divisible by b.
then 3c² is divisible by b.

This contradicts our assumption that root 3 rational.our assumption is wrong.

so √3 is irrational.