Here is one way to look at the problem (although, it doesn't use the extra information about q)
If p and q are both rational numbers, then so is p+q
So if you can show that
`root(3)(2)` is not rational, then you are done
You can do this in the "standard way" that you can show that `sqrt(2)` is not rational:
(see: http://en.wikipedia.org/wiki/Square_root_of_2#Proof_by_infinite_descent )
Suppose (by contradition) that `root(3)/(2)` were rational.
Then you could write:
`root(3)(2) = (a)/(b)` where a and b are natural numbers (whole numbers), and the fraction `(a)/(b)` is in lowest terms (it is "irreducible")
Cube both sides:
The left hand side is even, because it is divisible by 2. So the right hand side must be even as well. But then `a` must be even. (If you cube an odd number, the result is an odd number.) If `a` is even, we can write it as `a=2k` , where `k` is a whole number.
Then `a^3= (2k)^3 = 8k^3`
Now the right hand side is even. So the left must be even as well. But then `b` must be even (since an odd cubed is odd).
But then `a` and `b` are both even, and we said they were in lowest terms.
So `root(3)(2)` can't be written as ANY rational, and so it can't be written as a sum of rationals.