Rational exponents. Let be a positive integer, and let be a positive real number. Then we can find a unique value so that . We call such value the -th root of , and write `` .'' Furthermore, for any rational number we can define the rational exponents by
Irrational exponents. Let be an irrational number. Then, for a rational number arbitrarily close to we can find a unique value so that the rational exponent becomes arbitrarily close to . We call such value the irrational exponent .
Compound interest. Suppose that represents the principal, and that is an interest rate. The amount after interest period is expressed by
The number . We can find a unique value so that