Roots of higher powers

For any positive real number $a$ and positive integer $n$, the $n$th root of $a$ is the number $x$ such that $x^n = a$. We write $x = \sqrt[n]{a}$. Roots of higher powers generalize the idea of square and cube roots to any integer $n \geq 2$. Roots can also be written as exponents: $\sqrt[n]{a} = a^{1/n}$. For example, $\sqrt[5]{32} = 32^{1/5} = 2$.

Important properties

• The $n$th root of $a$ is $a^{1/n}$.

• $\sqrt[n]{a^m} = a^{m/n}$.

• If $n$ is even, $\sqrt[n]{a}$ is only defined for $a \geq 0$ (in real numbers).

• If $n$ is odd, $\sqrt[n]{a}$ is defined for all real $a$.

• $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ for $a, b \geq 0$.