Question

Prove that `(5 sqrt3 + 2/3)` is an irrational number given that `sqrt3` is an irrational number.

Answer 1

`sqrt3` is an irrational number, and we need to prove that `5sqrt3 + 2/3` is also irrational.

Assume `5sqrt3 + 2/3` is an irrational number

Let: `5sqrt3 + 2/3 = r`

where r is a rational number.

Rearrange the equation:

`5 sqrt3 = r - 2/3`

Since r and `2/3` are rational numbers, their difference `r - 2/3` is also a rational number.

`5sqrt3` = rational number

Dividing both sides by 5: 

`sqrt3 = (r - 2/3)/5`

Since the right-hand side is a rational number, this means `sqrt3` is rational. However, this contradicts the given fact that `sqrt3` is irrational.

Since our assumption that `5sqrt3 + 2/3` is rational leads to a contradiction, we conclude that:

`5sqrt3 + 2/3` is an irrational number.