## Powers and Roots.

For any positive integer , the function is continuous and strictly increasing on . Moreover and . Therefore is a bijection from onto itself. It is invertible and its inverse is called the root function.

We denote:

Remark 5.7.3   We give here a definition of roots valid only for non negative numbers. Many pocket calculators compute roots for negative numbers; in fact their definition'' of roots is different from what we explain here, and coincides with ours for non negative numbers.

Properties 5.7.4

1. .
2. .
3. .

We can now define rational powers of a non negative real number:

Definition 5.7.5   let be a rational number; we denote it in reduced form , where is an integer and is a positive integer. Then .

For example: .

Properties 5.7.6

1. .
2. .

For example:

.

Noah Dana-Picard 2007-12-28