So I know that it is possible for the sums and products of irrational numbers to be rational. But, the only instances I know of that happening is when a certain combination of additive or multiplicative inverses of the irrational numbers in question are used.
My question is, given an irrational number $p$, can you multiply or add an irrational number $q$ to it so that their sum/product is a rational number given that $q$ cannot be written as any combination involving either $-p$ or $p^{-1}$?
On
Let $a$ be irrational. If $b$ is some irrational number such that $a+b$ is rational then you may write $a+b=p/q$, where $p,q$ are integers. But then $b=p/q-a$.
Similarly, if $b$ is some irrational number such that $ab$ is rational then you may write $ab=p/q$, and so $b=p/q*a^-1$.
So, depending on precisely what you mean by "combination of ..." the above observations will probably give you an answer.
If $p+q = n/m$ then $q = n/m - p$ which is "a combination" involving $-p$. Same is true for the product.