Question

Show that 3+5root2 is irrational

Answer 1

Let 3+5√2 be a rational number.
A rational number can be written in the form of p/q where p,q are integers and q ≠ 0

3 + 5√2 = p/q
5√2 = p/q - 3
5√2 = (p - 3q)/q
√2 = (p - 3q)/5q

p,q are integers then (p - 3q)/5q is a rational number.
Then √2 must be a rational number.
But this contradicts the fact that √2 is an irrational number.
So, our supposition is false.
Hence, 3+5√2 is an irrational number.