Roots. Powers with rational exponents (other)

Rational exponents allow us to write roots and powers in a unified way. For any positive real number $a$, and rational number $r = \frac{m}{n}$ (where $m$ and $n$ are integers, $n > 0$), we define $a^{r} = a^{\frac{m}{n}} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$. This lets us work with expressions like $a^{3/4}$ or $a^{-2/3}$ using the rules of exponents.

Important properties

• For $a > 0$, $a^{\frac{m}{n}} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$.

• Negative exponents: $a^{-r} = 1/a^r$.

• Exponent rules still apply: $a^{p} \cdot a^{q} = a^{p+q}$, $(a^{p})^{q} = a^{pq}$.

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

• If $a < 0$ and $n$ is even, $\sqrt[n]{a}$ is not a real number.