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.

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

For any real number

a

and natural number

n,

the expression

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

is defined as the

n^{th}

root of

a .

Note that a root with an even index is defined only for non-negative numbers. Therefore, if

n

is even, then

a

must be non-negative.

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

Concept

$n^{th}$ Root

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