Let $x, \ r \in \mathbb{Q}$.
I need to find the conditions on $ \ x, \ r$ so that the value of $ \large (1+x)^r$ is a rational number.
Which $x, \ r$ makes $(1+x)^r$ a rational number?
Answer:
If I take $x=\frac{16}{9}$ and $r=\frac{1}{2}$, then $ (1+x)^r=(1+\frac{16}{9})^{\frac{1}{2}}=\sqrt{\frac{25}{9}}=\pm \frac{5}{3} \in \mathbb{Q}$,
If I take $x=\frac{19}{8}$ and $r=\frac{1}{3}$, then $ (1+x)^r=(1+\frac{19}{8})^{\frac{1}{3}}=\large \sqrt[3]{\frac{27}{8}}= \frac{3}{2} \in \mathbb{Q}$,
and so on $ \cdots $
How to find all $x \in \mathbb{Q}$ and $r \in \mathbb{Q}$ such that $(1+x)^r$ becomes a rational number?
Can you give me the general form of $x$ and $r$ so that $ (1+x)^r$ becomes a rational number?