Suppose we know that a3=8. We want to find what number raised to the 3rd power is equal to 8. Since 23=8, we say that 2 is the cube root of 8.
The nth root of a is a number that, when raised to the nth power, gives a. For example, −3 is the 5th root of −243 because (−3)5=−243. If a is a real number with at least one nth root, then the principal nth root of a is the number with the same sign as a that, when raised to the nth power, equals a.
The principal nth root of a is written as na, where n is a positive integer greater than or equal to 2. In the radical expression, n is called the index of the radical.
A General Note: Principal nth Root
If a is a real number with at least one nth root, then the principal nth root of a, written as na, is the number with the same sign as a that, when raised to the nth power, equals a. The index of the radical is n.
Example 10: Simplifying nth Roots
Simplify each of the following:
5−32
44⋅41,024
−31258x6
843−448
Solution
5−32=−2 because (−2)5=−32
First, express the product as a single radical expression. 44,096=8 because 84=4,096
3125−38x65−2x2Write as quotient of two radical expressions.Simplify.
843−243643Simplify to get equal radicands.Add.
Try It 10
Simplify.
3−216
453480
639,000+73576
Solution
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College Algebra.Provided by: OpenStaxAuthored by: OpenStax College Algebra.Located at: https://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1/Preface.License: CC BY: Attribution.