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Chapter 0: Problem 126

Describe the kinds of numbers that have rational fifth roots.

Short Answer

Expert verified
Rational numbers and their negative counterparts, as well as zero, have rational fifth roots, provided they can be expressed as the perfect fifth power of some rational number. This includes both integers and fractions where the denominator is not zero.

Step by step solution

01

Understanding Rational Numbers

Rational numbers are numbers that can be expressed as a fraction, where both the numerator and denominator are integers, and the denominator is not zero.
02

Perfect Fifth Power and Rational Fifth Roots

A number has a rational fifth root if that number can be expressed as a perfect fifth power of some rational number. In other words, if a number \(a\) can be written as \(b^5\) where \(b\) is a rational number, then \(a\) will have a rational fifth root.
03

Examples

For instance, \(32\) has a rational fifth root because it can be written as \((2)^5\), where \(2\) is a rational number. Similarly, \(1/(2^5)\) or \(0.03125\) also has a rational fifth root because it can be expressed as \((1/2)^5\), where \(1/2\) is a rational number.

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