Question

Show that `(4 + 3sqrt(2))` is irrational.

Answer 1

Given: `sqrt(2)` is irrational.

Step-wise calculation:

1. Suppose, for contradiction, that `4 + 3sqrt(2)` is rational.

Then there exist integers a, b (b ≠ 0) with `4 + 3sqrt(2) = a/b`.

2. Subtract 4 (rational) from both sides: `3sqrt(2) = a/b - 4`. 

The right-hand side is rational difference of rationals is rational.

3. Divide by 3 (nonzero rational):

`sqrt(2) = (a/b - 4)/3`

So, `sqrt(2)` would be rational.

4. This contradicts the fact that `sqrt(2)` is irrational.

The assumption that `4 + 3sqrt(2)` is rational leads to a contradiction; therefore `4 + 3sqrt(2)` is irrational.