nth root

The phrase ``the $n$ -th root of a number'' is a somewhat misleading concept that requires a fair amount of thought to make rigorous.

For $n$ a positive integer, we define an $n$ -th root of a number $x$ to be a number $y$ such that $y^n=x$ . The number $n$ is said to be the index of the root. Note that the term ``number'' here is ambiguous, as the discussion can apply in a variety of contexts (groups, rings, monoids, etc.) The purpose of this entry is specifically to deal with $n$ -th roots of real and complex numbers.

In an effort to give meaning to the term the $n$ -th root of a real number $x$ , we define it to be the unique real number that $y$ is an $n$ th root of $x$ and such that $\operatorname{sign}(x)=\operatorname{sign}(y)$ , if such a number exists. We denote this number by $\sqrt[n]{x}$ , or by $x^{\frac{1}{n}}$ if $x$ is positive. This specific $n$ th root is also called the principal $n$ th root.

Example: $\sqrt[4]{81} = 3$ because $3^4 = 3 \times 3 \times 3 \times 3= 81$ , and $3$ is the unique positive real number with this property.

Example: If $x+1$ is a positive real number, then we can write $\sqrt[5]{x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1} = x + 1$ because $(x + 1)^5 = (x^2 + 2x + 1)^2(x + 1) = x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1$ . (See the Binomial Theorem and Pascal's Triangle.)

The nth root operation is distributive for multiplication and division, but not for addition and subtraction. That is, $\sqrt[n]{x \times y} = \sqrt[n]{x} \times \sqrt[n]{y}$ , and $\sqrt[n]{\frac{x}{y}} = \frac{\sqrt[n]{x}}{\sqrt[n]{y}}$ . However, except in special cases, $\sqrt[n]{x + y} \not= \sqrt[n]{x} + \sqrt[n]{y}$ and $\sqrt[n]{x - y} \not= \sqrt[n]{x} - \sqrt[n]{y}$ .

Example: $\sqrt[4]{\frac{81}{625}} = \frac{3}{5}$ because $\left(\frac{3}{5}\right)^4 = \frac{3^4}{5^4} = \frac{81}{625}$ .

Note that when we restrict our attention to real numbers, expressions like $\sqrt{-3}$ are undefined. Thus, for a more full definition of $n$ th roots, we will have to incorporate the notion of complex numbers: The nth roots of a complex number $t = x + yi$ are all the complex numbers $z_1, z_2, \ldots, z_n \in \mathbb{C}$ that satisfy the condition $z_k^n = t$ . Applying the fundamental theorem of algebra (complex version) to the function $x^n-t$ tells us that $n$ such complex numbers always exist (counting multiplicity).

One of the more popular methods of finding these roots is through trigonometry and the geometry of complex numbers. For a complex number $z=x+iy$ , recall that we can put $z$ in polar form: $z=(r, \theta)$ , where $r = \sqrt[2]{x^2 + y^2}$ , and $\theta = \frac{\pi}{2}$ if $x = 0$ , and $\theta = \arctan{\frac{y}{x}}$ if $x \not= 0$ . (See the Pythagorean Theorem.) For the specific procedures involved, see calculating the nth roots of a complex number.