Question

Prove that the following is irrational:

`7sqrt5`

Answer 1

Let a = `7sqrt5` be a rational number.

We can find two co-prime integers a and b such that `7 sqrt5 = a/b` Where b ≠ 0

Now, `7 sqrt5 = a/b`

⇒ `a/(7b) = sqrt5.`

Now, `a/7` is a rational number since the product of two rational numbers is a rational number.

The above will imply that `sqrt5` is a rational number. But `sqrt5` is an irrational number.

∴ Our assumption is wrong.

Thus, we conclude that `7 sqrt5` is irrational.