Comparing two exponentiations

Question

$b^n$ where the base $b$ is a positive integer greater than $1$ and the exponent $n$ is a rational number in simplified form. How would one compare (resulting in <, =, or >) two such exponentiations without evaluating the exponentiatoins, and without the use of functions or operations that produce real numbers (e.g., log(), pow(), etc)?

Answer 1

In order to compare, say, $b_{1}^{p_{1}/q_{1}}$ and $b_2^{p_2/q_2}$, the simplest approach is to raise both to the same power ($q_1 q_2$) and compare the resulting integers, $b_{1}^{p_1 q_2}$ and $b_2^{p_2 q_1}$.