The Root of a Number

​The root |n^e| of a number |a| is denoted as follows. ||\sqrt[n]{a}|| ||\text{where}\ \ a\in \mathbb{R}\quad \text{and }\quad n \in \mathbb{N}^*|| A root |n^e| of a number |a| is a number that, when raised to the power of exponent |n|, equals |a|.

The sixth root of |64,| denoted |\sqrt[6]{64},| is |2| because |2^6=64.|

The fourth root of |81,| denoted |\sqrt[4]{81},| is |3| because |3^4=81.|

The radicand is the number, or algebraic expression, whose root is desired. In other words, it is the number or expression located inside the root symbol, also called the radical.

Meanwhile, the index is the numerical value directly associated with the root.

||\begin{align} &&&&& \color{red}{\text{radicand}} && = && \color{red}{8} \\
\sqrt[\color{blue}{3}]{\color{red}{8}}&= \color{magenta}{2} &&\large\Rightarrow && \color{blue}{\text{index}} && = && \color{blue}{3} \\
&&&&& \color{magenta}{\text{root}} && = && \color{magenta}{2} \end{align}||

The terminology for the symbol |\sqrt{\phantom{2}}| varies depending on the value of the index.

• When the index is |2,| it reads as "the square root" of the number. When the index is |2,| it is generally not written down.

• When the index is |3,| it reads as "the cube root" of the number.

• When the index is greater than |3,| add “th” to the end of the index value.

Example 1:

|\sqrt{16}| reads as "the square root of |16|" and has a value of |4|, since |4| to the power of |2| equals |16.| ||\sqrt{16}=\color{red}{4}\quad \Leftrightarrow \quad \color{red}{4}^{2}=16||
Example 2:

|\sqrt[3]{27}| reads as "the cube root of |27|" and has a value of |3|, since |3| to the power of |3| equals |27.| ||\sqrt[3]{27}=\color{red}{3}\quad \Leftrightarrow \quad \color{red}{3}^{3}=27||
Example 3:

|\sqrt[4]{625}| reads as "the fourth root of |625|” and has a value of |5|, since |5| to the power of |4| equals |625.| ||\sqrt[4]{625}=\color{red}{5}\quad \Leftrightarrow \quad \color{red}{5}^{4}=625||