Question

prove that (5 + 3 root 2 is an irational number

Answer 1

Let us assume that 5 + 3 root 2 to be rational.
5 + 3 root 2 = p/q (p and q are two completely prime positive integers)
3 root 2 = p/q -5
3 root 2 = p-5q/q
root 2 = p-5q / 3q
here, p, q, 5 and 3 and rational numbers. hence root 2 is rational number.
but we know that root 2 is irrational.
this contradiction has arisen due to our wrong assumption that 5 + 3 root 2 is rational.
So, we get that 5 + 3 root 2 is irrational.
hence proved.