I am stuck on the following example/problem in the book and am seeking clarification:
Page 76:
It is instructive to compare this result with the assertion that α = √2
and β = 2 log₂(3) yield α^β = 3 as Mathematica confirms. This illustrates
nicely that verification is often easier than discovery. Similarly, the fact that
multiplication is easier than factorization is at the base of secure encryption
schemes for e-commerce.
Indeed, there are eight possible rational/irrational triples: α^β = γ;
a few pages down, the following problem appears:
28. Eight solutions. Find examples of all eight rational and irrational
possibilities of α^β = γ
I am not too sure what is being said by these "8 possibilities", for, there are surely much more than 8 examples of the precise form $\sqrt{p}^{q \log_b{a}}$.
The book does not have accompanying solutions as far as I am aware, so I do not have any pointers to try looking into.
Here's the intent of the exercise . . .
Consider the equation $\alpha^\beta=\gamma$, where each of $\alpha,\beta,\gamma$ can be rational or irrational.
Of those $2^3=8$ cases, give an example of each.