Question

Prove that `2 + 3sqrt(3)` is an irrational number when it is given that `sqrt(3)` is an irrational number.

Answer 1

To prove: `2 + 3sqrt(3)` is irrational, let us assume that `2 + 3sqrt(3)` is rational.

`2 + 3sqrt(3) = a/b; b ≠ 0` and a and b are integers.

⇒ `2b + 3sqrt(3)b = a`

⇒ `3sqrt(3)b = a - 2b`

⇒ `sqrt(3) = (a - 2b)/(3b)`

Since a and b are integers so, `a - 2b` will also be an integer.

So,`(a - 2b)/(3b)` will be rational which means `sqrt(3)` is also rational.

But we know `sqrt(3)` is irrational (given).

Thus, a contradiction has risen because of an incorrect assumption. 

Thus, `2 + 3sqrt(3)` is irrational.