Lesson 1: nth Roots 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

If $b^n = a$, then $b$ is an $n^{\text{th}}$ root of $a$.
The principal $n^{\text{th}}$ root of $a$ is written $\sqrt[n]{a}$.
$n$ is called the index of the radical.
Properties of $\sqrt[n]{a}$
When $n$ is an even number and:

• $a \geq 0$, then $\sqrt[n]{a}$ is a real number.
• $a < 0$, then $\sqrt[n]{a}$ is not a real number.

When $n$ is an odd number, $\sqrt[n]{a}$ is a real number for all values of $a$.

Examples

• To simplify $\sqrt[3]{125}$, we look for a number that, when cubed, is 125. Since $5^3 = 125$, $\sqrt[3]{125} = 5$.