Algebra · Properties of exponents
You were handed eight of them to memorise. Seven are consequences of the first one refusing to break.
What you’ve derived so far
The ledger
An exponent starts life as shorthand. a3 means three copies of a multiplied together — nothing more. Each tile below is one copy.
How many copies of a are multiplied in a0? None. So the tiles say the answer is nothing at all — zero.
Let’s test that against the rule you just proved.
Multiplying something by itself negative once is not a thing you can do. But the law does not care what you can picture. It only cares about being obeyed.
This time the product rule is not the one doing the forcing. The power rule is.
Nothing new needs deciding here. Every move below is a rule you have already unlocked. Name the one that justifies each line.
Four moves, and one rule gets used twice.
Only the top row was ever a discovery. Everything below it was forced: at each step there was exactly one value that let the law survive, and that value became the definition. Nobody voted on a0 = 1. It was the only option left standing.
Two conditions on the printed table finally make sense.
Most textbook tables print “a ≠ 0” above the whole list, and for the first seven rules that is enough. Fractional exponents need more. (−8)1/3 = −2 is perfectly fine, but (−4)1/2 asks for a number that squares to −4, and no real number does. Once you allow arbitrary fractional exponents on a negative base, the law you spent this lesson protecting stops holding. So the honest condition for the last two rows is a > 0.