Problem 126 - College Algebra

Question

Describe the kinds of numbers that have rational fifth roots.

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: Conclusion

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