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Chapter 0: Problem 126

Describe the kinds of numbers that have rational fifth roots.

Short Answer

Expert verified
Numbers that are a perfect fifth power of a rational number have rational fifth roots.

Step by step solution

01

Understand Rational Numbers

Rational numbers are numbers that can be expressed as a ratio or a division of two integers, where the denominator is not zero. They include whole numbers, fractions, decimal numbers that repeat or terminate.
02

Understand Rational Roots

Roots of a number are the values that, when raised to a certain power, result that number. A root is rational if it can be expressed as a rational number. For nth roots (such as a fifth root which is a 1/5th root), if a rational number exists that when raised to the power of n gives the given number, then the given number has a rational nth root.
03

Understand Rational Fifth Roots

Numbers that have rational fifth roots are those for which there is a rational number that, when raised to the power of 5, equals the given number. To enable a radical to produce a rational number, the radicand must be a perfect fifth power of a rational number.
04

Conclusion

So, the kinds of numbers that have rational fifth roots are those that are a perfect fifth power of a rational number.

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