- In the radical , the index is 2, and the radicand is . It is read as "the square root of the quantity plus 2."
- In the radical , the index is 3, and the radicand is . It is read as "the cube root of squared."
The rules for multiplying or dividing radicals are the same as with exponents. A product or quotient under a radical can be split into two separate radicals, and radicals with the same index that are multiplied or divided can be combined into one.
Properties of radicals are used to simplify radical expressions. It is also possible to combine like radicals, which have the same radicand and index.
Multiplying and Dividing Radicals
|Product of radicals property||The radical of a product is the product of the radicals.|
|Quotient of radicals property||The radical of a quotient is the quotient of the radicals.|
Simplest Radical Form
Rules for Simplifying Radicals
|There are no perfect th powers under a radical of index , such as perfect squares within a square root or perfect cubes under a cube root.|
|There are no fractions under a radical or radicals in the denominator of a fraction.|
Rationalizing the Denominator
Rationalizing the denominator is the process of rewriting a fraction to remove radical expressions from the denominator. To rationalize the denominator, multiply the fraction by a fraction that is equivalent to 1. The type of radical expression in the denominator determines which fraction equivalent to 1 should be used.If the denominator has two terms, such as , the conjugate of the expression is . When two expressions are multiplied, the resulting expression does not contain a radical:
How to Rationalize Denominators
|Denominator||Fraction Equivalent to 1||Example|
|Single square root||Multiply by a fraction in this form:
|Single cube root||Multiply by a fraction in this form:
|Expression with two terms, such as:
||Multiply by a fraction in this form: