In Exercises 59–76, find the indicated root, or state that the expression is not a real number. ___ ⁵√−1

Verified step by step guidance

1. Recognize that you are asked to find the fifth root of \(-1\).
2. Recall that the nth root of a number is a value that, when raised to the power of n, gives the original number.
3. Consider the properties of real numbers and roots: the nth root of a negative number is not a real number if n is even, but it can be real if n is odd.
4. Since 5 is an odd number, the fifth root of \(-1\) is a real number.
5. Identify that the fifth root of \(-1\) is a number that, when raised to the power of 5, equals \(-1\).

Key Concepts

Roots of Numbers

Roots of numbers refer to the operation of finding a value that, when raised to a certain power, yields the original number. For example, the square root of 9 is 3 because 3² = 9. In this case, we are dealing with the fifth root, which means we are looking for a number that, when multiplied by itself five times, equals -1.

Odd vs. Even Roots

Odd roots, such as the fifth root, can yield real numbers for negative inputs. This contrasts with even roots, like square roots, which do not produce real results for negative numbers. Therefore, while the square root of -1 is not a real number, the fifth root of -1 is a real number, specifically -1, since (-1)⁵ = -1.

Real Numbers

Real numbers include all the numbers that can be found on the number line, encompassing both rational and irrational numbers. They include positive numbers, negative numbers, and zero. Understanding whether a number is real is crucial when evaluating roots, especially when dealing with negative values and their respective roots.