Radical & Rational Exponent

The principal nth root of a is written as ${\sqrt[{n}]{a}}$, where n is a positive integer greater than or equal to 2. In the radical expression, n is called the index of the radical. Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer, and if the index n is even, then a cannot be negative.

${\sqrt[{n}]{a}}=a^{\frac {1}{n}}$

We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an nth root. The numerator tells us the power and the denominator tells us the root.

$a^{\frac {m}{n}}=({\sqrt[{n}]{a}})^{m}={\sqrt[{n}]{a^{m}}}$

All of the properties of exponents that we learned for integer exponents also hold for rational exponents.

Resources

• Radicals and rational exponents - Basic example, Khan Academy
• Radicals and Rational Exponents, OpenStax

Licensing

Content obtained and/or adapted from:

• Radicals and Rational Exponents, OpenStax College Algebra under a CC BY-SA license