root

$n$-th root

For $x\ge 0$ and $n\in \mathbb{N}$:

$$\sqrt[n]{x}=x^{1/n}=y⟺y^n=x\text{ and }y\ge 0$$

$\sqrt{x^2}=|x|$

Not $x$, because $x$ might be negative. E.g. $\sqrt{(-3{)}^2}=\sqrt{9}=3=|-3|$.

Properties

For $a,b\ge 0$:

$$\sqrt[n]{a\cdot b}=\sqrt[n]{a}\cdot \sqrt[n]{b},\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}(b\ne 0)$$

In particular, dividing “through” a root:

$$\frac{\sqrt{x}}{a}=\frac{\sqrt{x}}{\sqrt{a^2}}=\sqrt{\frac{x}{a^2}}(a>0)$$

This is useful for simplifying limits: e.g. $\frac{\sqrt{n^2+2n}}{n}=\sqrt{\frac{n^2+2n}{n^2}}=\sqrt{1+\frac{2}{n}}$