Problem 115 - College Algebra

Question

Describe the kinds of numbers that have rational fifth roots.

Step-by-Step Solution

Answer

The kinds of numbers that have rational fifth roots are those which are the fifth powers of any positive or negative integer, as well as zero. This set of numbers make up the rational fifth roots.

Step 1: Understanding the Concept of Roots

A root of a number refers to a value that, when raised to a certain power, yields that number. For example, if \(x^5 = a\), then \(x\) is a fifth root of \(a\). Therefore, the concept of a fifth root can be understood as an operation that finds the initial value which, when multiplied by itself five times, will provide the desired number. In the context of this exercise, we seek a fifth root that is a rational number, meaning it can be written as a fraction.

Step 2: Category of Numbers

Rational numbers have the ability to be expressed as fractions that include both positive and negative integers, including zero. If we have a number that under the operation of a fifth root gives a rational number, this means that this number can be the fifth power of an integer (positive or negative) or zero. Any number that is not within these categories would not provide a rational fifth root.

Step 3: Deducing the Numbers

From the criteria defined in Step 2, it can be concluded that rational fifth roots include the fifth powers of any positive or negative whole number, as well as zero. This is due to the fact that fifth roots of these numbers can always be expressed as fractions (thus classifying them as rational).