Question

6 under root 5 is irrational number prove it

Answer 1

Let assume 6√5 is a rational number
6√5= p/q
√5=p/6q

Now again,
√5= a/b

on squaring to both the sides
(√5)^2= (a/b)^2
5= a^2/b^2
5b^2= a^2
b^2= a^2/5

By theorem 1.3, it divides a^2 so it will divides a also

on putting a= 5c

b^2= (5c)^2/5
b^2= 25c^2/5
b^2= 5c^2
b^2/5= c^2

Again, by theorem 1.3, it divides b^2 so it will divides b also

As we can see that the common factor is 5

Hence, our assumption is wrong as 6√5 is rational number

Therefore, 6√5 is an irrational number

HOPE it helps!!!!