Problem 126 - College Algebra Essentials

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 rational numbers.

Step 1: Definition of Rational Numbers

Rational numbers are defined as numbers that can be expressed as the quotient or fraction \(p/q\) of two integers, with the denominator \(q\) not equal to zero. These include integers themselves (as they can be expressed with a denominator of 1), fractions, decimals that terminate or repeat, among others.

Step 2: Understanding Roots

When we calculate the fifth root of a number, we are essentially asking the question: 'Which number, when multiplied by itself four more times (i.e. raised to the power of 5), gives us our original number?' This operation is the inverse of raising a number to the power of 5.

Step 3: Identify Types of Numbers with Rational Fifth Roots

Any rational number, when raised to the power of 5, would result in a rational number. The fifth root of this result (which is a rational number) would then be the original number, which is rational. Therefore, the types of numbers that have rational fifth roots are rational numbers.