nth root. Properties of the nth root

Arithmetic root of degree

n

Let

n\in\mathbb{N}

, $n\ge 2$ , and $a\ge 0$ . The arithmetic root of degree $n$ of

a

is the non-negative number

b

such that $b^n=a$ . Notation: $\sqrt[n]{a}$ (“the root of degree

n

a

”). For

a>0

such

b

exists and is unique. Cases

a=0

and

a=1

fit intuition: $\sqrt[n]{0}=0$ , $\sqrt[n]{1}=1$ . For $n=2$ one writes $\sqrt{a}$ (square root); for $n=3$ — cube root

\sqrt[3]{a}

The word “arithmetic” stresses choosing the non-negative

b

when

a\ge 0

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