Concept

$n^{th}$ Root

The $n^{th}$ root of a number $a$ expresses another number that, when multiplied by itself $n$ times, will result in $a .$ Aside from the radical symbol, the notation is made up of the radicand $a$ and the index $n .$

The resulting number is commonly called a radical. For example, the radical expression $416$ is the fourth root of $16 .$ $416$ simplifies to $2$ because $2$ multiplied by itself $4$ times equals $16 .$ $416 = 4 2^{4} = 2$ The general expression $n a$ represents a number which equals $a$ when multiplied by itself $n$ times.

$n times n a \cdot n a \cdot \dots \cdot n a = a or (n a)^{n} = a$

The $n^{th}$ root can also be expressed using a rational exponent.

$n a^{1} = a^{\frac{1}{n}}$

Just as with exponents, the most common roots have special names: square roots and cube roots have an index of $2$ and $3,$ respectively.