4.7 Roots

Let us now consider a real number $a$ and a natural number $n$. We use powers with natural exponents to define natural roots as a certain (see below) real solution of the equation $x^n=a$. We then symbolically denote this solution in the following two ways

\begin{equation*} a^{\frac{1}{n}} = \sqrt[n]{a}, \quad n\in\mathbb{N}. \end{equation*}

It is necessary to distinguish between the cases of odd and even $n$ and to consider whether such a construction makes sense or not.

4.7.1 Even roots

If it is $n=2k$, $k\in\mathbb{N}$, thus $n$ is even, then $x^n\geq 0$ for every $x\in\mathbb{R}$. This means that the equation $x^n = a$ has a real solution only for $a\geq 0$. This situation is shown in figure 4.5 for $n=2$. For $a>0$ the solutions of such equation are actullay two, since $x^{2k}=(-x)^{2k}$.

Figure 4.5: Construction of the square root of a number $a$.

We define the even root $\sqrt[2k]{a}$ as the non-negative solution of the equation $x^{2k} = a$. Therefore, for example, $\sqrt{x^2}$ is $|x|$ and not $x$. For $a=0$ the solution is just one and it is $\sqrt[2k]{0} = 0$.

From the above discussion, it is clear that the domain and the image set of the function $f(x)=\sqrt[2k]{x}$ are both given by the set $[0, +\infty)$. Furthermore, the following equality applies

\begin{equation*} \sqrt[2k]{x^{2k}} = \left(\sqrt[2k]{x}\right)^{2k} = x \quad \text{for every} \ x \geq 0. \end{equation*}

In other words, $\sqrt[2k]{x}$ is the inverse function of $x^{2k}$ restricted to the set $[0,+\infty)$. See picture 4.6 for $k=1$. This will be discussed in more detail in BIE-ZMA.

Figure 4.6: Square power and square root.

4.7.2 Odd roots

If it is $n=2k-1,k\in\mathbb{N}$, thus $n$ is odd, then the equation $x^{2k-1} = a$ has only one solution, that we call odd power of $a$ and we denote it as $\sqrt[2k-1]{a}$. For example, $\sqrt[3]{-8}=-2$. See picture 4.7 for the case $n=3$.

Figure 4.7: Construction of the cubic root of the number $a$.

The domain and image set of an odd root are both given by the whole real line $\mathbb{R}$. An odd power and the corresponding odd root are inverse to each other, namely it holds

\begin{equation*} \sqrt[2k-1]{x^{2k-1}} = \left(\sqrt[2k-1]{x}\right)^{2k-1} = x \quad \text{for every} \ x \in \mathbb{R}. \end{equation*}

For illustration in the case $k=2$, see picture 4.8.

Figure 4.8: Cubic power and cubic root.