Question

Given that `sqrt(3)` is an irrational number, show that `(5 + 2sqrt(3))` is an irrational number.

Answer 1

Let us assume `(5 + 2sqrt(3))` is a rational number.

∴ `5 + 2sqrt(3) = p/q`

(where, q ≠ 0 and p and q are coprime integers)

⇒ `sqrt(3) = (p - 5q)/(2q)`

This contradicts the given fact that `sqrt(3)` is irrational.

Hence, `(5 + 2sqrt(3))` is an irrational number.