Arithmetic Root of the n-th Degree
A comprehensive guide to understanding, simplifying, and manipulating n-th roots
1. Definition
Definition: The arithmetic n-th root of a non-negative number $a$ is the unique non-negative number $b$ such that $b^n = a$. We write:
$$ b = \sqrt[n]{a} $$
where $n \in \mathbb{N}$, $n \geq 2$, and $a \geq 0$.
Key requirements:
- The radicand must be non-negative: $a \geq 0$
- The result is always non-negative: $\sqrt[n]{a} \geq 0$
- $n$ is a natural number called the index (or degree) of the root
Examples:
- $\sqrt[2]{9} = 3$ because $3 \geq 0$ and $3^2 = 9$
- $\sqrt[3]{8} = 2$ because $2 \geq 0$ and $2^3 = 8$
- $\sqrt[4]{81} = 3$ because $3 \geq 0$ and $3^4 = 81$
- $\sqrt[5]{32} = 2$ because $2 \geq 0$ and $2^5 = 32$
⚠ Important:
The arithmetic root is always non-negative by definition. For example:
- $\sqrt[2]{4} = 2$, not $\pm 2$
- $\sqrt[4]{16} = 2$, not $\pm 2$
Even though $(-2)^2 = 4$, the arithmetic square root of 4 is only $+2$.
Interactive visualization — available in the full lesson
Graph: As the index $n$ increases, the root function grows more slowly for $x > 1$ and faster for $0 < x < 1$.