Hi,
Prove or disprove that
a. sum of two distinct irrational numbers is irrational.
b. Product of two distinct irrational numbers is irrational
I've always liked this proof that an irrational to an irrational power need not be irrational: consider $\displaystyle \sqrt2^{\sqrt2}$. If that's rational, then we have our counterexample; otherwise, a counterexample is $\displaystyle \left(\sqrt2^{\sqrt2}\right)^{\sqrt2}={\sqrt{}2}^{ \sqrt{2}\sqrt{2}}=\sqrt2^2=2$.