Question

Show that `5 +sqrt 7`  is an irrational number.

Answer 1

Let us assume that `5 +sqrt 7` is a rational number.

⇒ `5 +sqrt 7 = p/q`, where p and q are two integers and q ≠ 0

⇒ `sqrt 7 = p/q - 5 = (p-5q)/q`

Since, p, q and 5 are integers, so `(p - 5q)/ q` is a rational number.

But this contradicts the fact that `sqrt 7` is an irrational number.

This contradiction has arisen due to our assumption that  `5 +sqrt 7` is a rational number. 

Hence, `5 +sqrt 7` is an irrational number.