Roots. Powers with rational exponents

Roots and powers with rational exponents are closely related. For any positive real number $a$ and positive integer $n$, the $n$th root of $a$ is written as $a^{1/n}$, which is the number $x$ such that $x^n = a$. More generally, for any rational exponent $m/n$, $a^{m/n} = (a^{1/n})^m = \sqrt[n]{a^m}$.

Important properties

• For $a > 0$, $a^{1/n}$ is the unique positive real number whose $n$th power is $a$.

• For any rational exponent $m/n$, $a^{m/n} = (a^{1/n})^m = (a^m)^{1/n}$.

• Exponent rules extend to rational exponents: $a^{p/q} \cdot a^{r/s} = a^{p/q + r/s}$, $(a^{p/q})^{r/s} = a^{(p/q)\cdot(r/s)}$.

• Roots can be written as fractional exponents: $\sqrt[n]{a} = a^{1/n}$.