Radical & Rational Exponent
The principal nth root of a is written as , 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.
![{\displaystyle {\sqrt[{n}]{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7873203eb76042fcd24056c553de8c86054a2df)
![{\displaystyle {\sqrt[{n}]{a}}=a^{\frac {1}{n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2211e22214f87e6a2303d3d4c2465a3229984bde)
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.
![{\displaystyle a^{\frac {m}{n}}=({\sqrt[{n}]{a}})^{m}={\sqrt[{n}]{a^{m}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b357cc21513a3160898040ac2b78edbd28677177)
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
