## Rational and irrational exponents

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

and

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

 (1)

Department of Mathematics