Question

Prove that `7 + 4sqrt(5)` is an irrational number, given that `sqrt(5)` is an irrational number.

Answer 1

Let `7 + 4sqrt(5)` is a rational number

∴ `7 + 4sqrt(5) = p/q`  ...[where p and q are integers and q ≠ 0]

`7q + 4sqrt(5)q` = p

`sqrt(5) = (p - 7q)/(4q)`

p and q are integers

∴ `(p - 7q)/(4q)` is a rational no. while

`sqrt(5)` is an irrational number

So `sqrt(5) ≠ (p - 7q)/(4q)`

Hence our assumption is wrong

So `7 + 4sqrt(5)` is an irrational number by contradiction fact.