Question

Prove that `7 - 3sqrt5` is an irrational number, given that `sqrt5` is an irrational number.

Answer 1

Let us assume, to the contrary, that `7 - 3sqrt5` is rational

`7 - 3sqrt5 = "p"/"q"` where p and q are co-primes and q ≠ 0

`sqrt5 = ("p" - 7"q")/(-3"q")`

`sqrt5 = (7"q" - "p")/(3"q")`

Since p and q are integers

∴ `(7"q" - "p")/(3"q")` is a rational number

∴ `sqrt5` is a rational number, which is a contradiction as `sqrt5` is an irrational number.

Hence, our assumption is wrong and hence `7 - 3sqrt5` is an irrational number.