Problem 126 - Algebra and Trigonometry

Question

Describe the kinds of numbers that have rational fifth roots.

Step-by-Step Solution

Answer

Positive rational numbers, negative rational numbers, and zero have rational fifth roots, while irrational numbers generally do not.

Step 1: Analyzing Positive Rational Numbers

The fifth root of a positive rational number is a number, which when raised to the power of five, results in the original number. Therefore, any positive rational number whose fifth power is also rational, has a rational fifth root.

Step 2: Analyzing Negative Rational Numbers

The fifth root of a negative rational number is a number, which when raised to the power of five, results in the original number. Since raising any real number to an odd power, including 5, will preserve the sign of the base, negative rational numbers also have rational fifth roots.

Step 3: Analyzing Zero

Zero is the fifth root of itself, as any exponent of zero, except for zero itself, equals zero. Therefore, zero also has a rational fifth root.

Step 4: Analyzing Irrational Numbers

Irrational numbers are numbers that cannot be expressed as the fraction of two integer numbers. The fifth root of an irrational number is generally an irrational number, so irrational numbers generally do not have rational fifth roots.