Lesson 1: Rational Exponents and Properties of Exponents

Property

If $\sqrt[n]{a}$ is a real number and $n \ge 2$, $a^{\frac{1}{n}} = \sqrt[n]{a}$.
Rational exponents are another way of writing expressions with radicals.

Examples

• To write $b^{\frac{1}{5}}$ as a radical, the denominator 5 becomes the index of the root, so we get $\sqrt[5]{b}$.

• To simplify $81^{\frac{1}{4}}$, we rewrite it as $\sqrt[4]{81}$. Since $3^4 = 81$, the answer is $3$.

Property

For any positive integers $m$ and $n$,
$a^{\frac{m}{n}} = (a^{\frac{1}{n}})^m = (\sqrt[n]{a})^m$
and
$a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}} = \sqrt[n]{a^m}$
To simplify, it is usually best to take the root first to keep the numbers in the radicand smaller.

Examples

• To simplify $27^{\frac{2}{3}}$, we can write it as $(\sqrt[3]{27})^2$. The cube root of 27 is 3, and $3^2$ is $9$.

• To simplify $16^{\frac{5}{4}}$, we write it as $(\sqrt[4]{16})^5$. The fourth root of 16 is 2, and $2^5$ is $32$.