Exponentiation rules in \(\mathbb{R}\)
This is something one usually learns in a piece-wise fashion, because it is information that is usually scattered among all sorts of different books (analysis, algebra, etc.). Well, here I round it all up in one place.
→ [preprint] ← [Update history]
The text begins with the raising of real numbers to positive integers. From there, it moves on to negative integers, rational numbers, and finally real numbers, as exponents. The final part—real numbers as exponents—requires quite a bit of work with sequences, in particular computing their limits. The reader is assumed to be familiarised with that topic, which is usually taught in freshman calculus courses.
March 22, 2024. Got feedback? Great, email is your friend!
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Update history
April 11, 2024. In section 4 (real exponents), I added a note about division—similarly to that which exists at the end of sections 2 and 3. I also explicitly indicate now in corollary 4.9 that it implies that \(x^q\) is always positive. And I indicated that corollary 4.10 is present there just for completeness—it is not used anywhere else. The preprint version prior to these changes is here.