Many beginning students think it's weird that anything raised to the power of is . (I hear a lot of people say "It should be ") You can use the product of powers property to show why this must be true.
We know . So, this says that . What number times equals ? If we try , we have . No good. The only number that works is .
In general, for all real numbers a, , we have:
(Click here to see why.)