Hi Mellie is is great to see you on the forum again :))
Both guests have given great answers. Thanks guests.
I will assume that you are only talking about irrational numbers where the irational bit is a surd.
(a) Give an example of two irrational numbers which, when added, produce a rational number.
\(\sqrt5 \quad and \quad(-\sqrt5)\\ 3+\sqrt2 \quad and \quad 3-\sqrt2\\ \)
Any two numbers where the one has has the negative radical of the other one.
That is not very mathematically precise but I expect you know what I mean.
Now let's consider just the addition of radicals.
(b) Suppose that a and b are positive integers such that both sqrt a and sqrt b are irrational. For what values of and is rational? Prove your answer. None I think.....
(c) Again assuming a and b positive integers such that both sqrt a and sqrt b are irrational, for what values of and is rational? Prove your answer.
Lets see.
First you should fully understand this proof.
Prove sqrt(2) is irrational.
https://www.khanacademy.org/math/algebra/rational-and-irrational-numbers/proofs-concerning-irrational-numbers/v/proof-that-square-root-of-2-is-irrational
I am not yet sure how to expand upon this to prove what you need. ......
I am still thinking......
If you message CPhill, Alan, Heureka or Gino3141 they may be able to better help you.