Is the sum of rational exponentials a rational exponential?.

Question

Prove or disprove that $\forall a,b \in \mathbb{Q}^+$ and $ \forall p,q \in \mathbb{Q}$ there exists $c \in \mathbb{Q}^+$ and $r \in \mathbb{Q}$ such that: $$ a^p+b^q=c^r $$

Answer 1

Hint: Suppose that $\sqrt{1}+\sqrt{2}$ is of the shape $c^r$, where $c$ is a positive rational and $r=\frac{m}{n}$ where $m$ and $n$ are integers, with $n\gt 0$. Show by taking the $n$-th power of both sides that this implies that $\sqrt{2}$ is rational.