Question

Proove that 3+(5 root 2) is an irrational number.

Answer 1

let 3+5 root 2 is a rational no.
so, 3+5root2 = p/q
      5root2 = p/q-3
      root2= p/q-3 / 5
we can observe that p/q-3 / 5 is a rational no.________(1)

sim, root 2 = p/q as root 2 is also a rational no.
squaring both sides
we get,
2=p^2/q^2
2q^2=p^2
we can observe that 2 can divide p^2
now, put p=2m
2q^2 = (2m)^2
q^2=2m
we can observe that 2 can divide q^2________(2)

from (1) and (2)
3+5root2 is a rational no.

our supposition is wrong, its contradiction
hence 3+5root2 is an irrational no.