



 
When people first devised the power notation, $x^n$ was defined only for $n = 2,3,4,\dots$ meaning $n\; x's$ multiplied together. From that definition certain rules were obtained, such as $a^n \times b^n = (ab)^n,$ and Extending this to defining $x^1 = x$ was straightforward, even though there is no multiplication involved. Using this definition, all the rules still work. Clearly $a^1 \times b^1 = ab = (ab)^1;$ and more importantly $\begin{eqnarray}a^1 \times a^n & = & a \times (a \times a \times \dots \times a, \mbox{ n times })\\ Now we get curious and ask about $x^0.$ We need $a^0 \times a^n = a^{0 \; + \; n} = a^n.$ This forces Using the same idea  extend the definition while keeping the same properties  we can define negative powers (which are usually fractions), fractional powers (which are usually irrational and sometimes complex) and irrational and complex powers. We end up with strange and wonderful results like $e^{\pi i} = 1$ (where e is the base of the natural logarithms and i is the square root of 1)! Good Hunting!  


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. 