Lesson 1: nth Roots, Radicals, and Rational Exponents

Property

$s$ is called an $n$th root of $b$ if $s^n = b$. We use the symbol $\sqrt[n]{b}$ to denote the $n$th root of $b$. An expression of the form $\sqrt[n]{b}$ is called a radical, $b$ is called the radicand, and $n$ is called the index of the radical.

Examples

• $\sqrt[3]{27} = 3$ because $3^3 = 27$.

• $\sqrt[4]{256} = 4$ because $4^4 = 256$.

Property

For any positive integers $m$ and $n$,

It is usually easier to take the root first: $(\sqrt[n]{a})^m$. This keeps the numbers in the radicand smaller.

Examples

• To write $\sqrt[4]{z^3}$ with a rational exponent, the index 4 is the denominator and the power 3 is the numerator: $z^{\frac{3}{4}}$.

• To simplify $27^{\frac{2}{3}}$, take the cube root first: $(\sqrt[3]{27})^2 = (3)^2 = 9$.