[ Rational and irrational numbers ]

a) Can the sum of two rational numbers be irrational?

b) Can the sum of two irrational numbers be rational?

c) Can an irrational number raised to an irrational power be rational?
a) no. b) $\sqrt{2}+(-\sqrt{2})=0$.

c) Define the number $\alpha$ by the equation $(\sqrt{3})^{\alpha}=2$. If $\alpha=\frac{p}{q}$, then we obtain the impossible equation $3^{\mathrm{p}}=4^{\mathrm{q}}$.

Another example can be constructed using the number $\beta=\sqrt{ } 2^{\sqrt{ } 2}$. If it is rational, then the problem is solved. If

it is irrational, then $\beta^{\sqrt{2}}=2$ and the desired pair of numbers will be $\beta$ and $\sqrt{2}$.