Powers and Roots.

For any positive integer $q$ , the function $f_q: x \mapsto x^q$ is continuous and strictly increasing on $[0,+\infty )$ . Moreover $f_q(0)=0$ and $\underset{x \rightarrow -\infty}{\text{lim}} x^q = +\infty$ . Therefore $f_q$ is a bijection from $[0,+\infty )$ onto itself. It is invertible and its inverse $f_q^{-1}$ is called the $q^{th}-$ root function.

We denote:

$$\begin{cases}y=\sqrt[q]{x} x \geq 0 \end{cases} \Longleftrightarrow \begin{cases}x=y^q y \geq 0 \end{cases}$$

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. $\forall q \in \mathbb{N}-{0}, \forall x \in [0,+\infty), \forall y \in [0,+\infty), \; \sqrt[q]{xy}=\sqrt[q]{x} \cdot \sqrt[q]{y}$ .
2. $\forall q \in \mathbb{N}-{0}, \forall x \in [0,+\infty), \forall y \in (0,+\infty), \; \sqrt[q]{\frac xy}=\frac{\sqrt[q]{x}}{\sqrt[q]{y}}$ .
3. $\forall p \in \mathbb{N}-{0},\forall q \in \mathbb{N}-{0}, \forall x \in [0,+\infty), \; \sqrt[p]{ \sqrt[q]{x}}=\; \sqrt[pq]{x}$ .

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

Definition 5.7.5   let $\alpha$ be a rational number; we denote it in reduced form $\alpha = \frac pq$ , where $p$ is an integer and $q$ is a positive integer. Then $\forall x \in [0,+\infty ), \; x^{\alpha} = \sqrt[q]{x^p}$ .

For example: $8^{4/3}=(\sqrt[3]{8})^4=2^4=16$ .

Properties 5.7.6       

1. $\forall \alpha \in \mathbb{Q}, \forall x \in [0,+\infty), \forall y \in [0,+\infty), \quad x^{\alpha} \cdot y^{\alpha} =(xy)^{\alpha}$ .
2. $\forall \alpha \in \mathbb{Q}, \forall \beta \in \mathbb{Q}, \forall x \in [0,+\infty), \quad x^{\alpha} \cdot x^{\beta} = x^{\alpha + \beta}$ .

For example:

$25^{5/2} \cdot 125^{2/5}=(5^2)^{5/2} \cdot (5^3)^{5/2} =5^5 \cdot 5^{15/2} = 5^{5+15/2}=5^{25/2}=\sqrt{5^{25}}=5^{12}\sqrt{5}$ .