rational^rational…

(FB/Math/BM – Problem raised by Souradeep Purkayastha)

Which of the following do always hold?

rational ^ rational = rational
rational ^ rational = irrational
rational ^ irrational = rational
…rational ^ irrational = irrational
irrational ^ rational = rational
irrational ^ rational = irrational
irrational ^ irrational = rational
irrational ^ irrational = irrational
_________________________________________________

All eight equalities hold in some case from which it also follows that none holds in all cases.  Examples where they hold (example for one means counterexample for another):

rational ^ rational = rational: 2³, (1/8)1/3
rational ^ r…ational = irrational: 21/2
rational ^ irrational = rational: 10lg 2
rational ^ irrational = irrational: 2√2
irrational ^ rational = rational: (√2)2
irrational ^ rational = irrational: (√2)1/2
irrational ^ irrational = rational: (2√2)√2
irrational ^ irrational = irrational: (√2)√2

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