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Let $A:= \{1,2,3,\dots\}$ and $P := A \times A$. Now define a relation on $R$ on $P\,$ by

$$(x,y)R(a,b) \iff x^y = a^b$$

1) Determine the equivalence class $[(9,2)]$ of $(9,2)$

Note that I already verify it is indeed an equivalence relation.

But what to do with the points?

$9^2 = 81 = a^b$

Not sure what to do

2) Determine an equivalence class with exactly four elements

Need to do on question 1 first

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How many integer solutions are there to $a^b = 81 = 3^4$? Think about the Fundamental Theorem of Arithmetic (unique factorization into primes).

For (2), again, think about the Fundamental Theorem of Arithmetic, and what you need to be able to write $a^b$ in exactly four different ways.

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