In Exercises 59–76, find the indicated root, or state that the expression is not a real number. ___ ⁶√−1

Verified step by step guidance

1. Identify the expression: \( \sqrt[6]{-1} \).
2. Recognize that you are looking for the sixth root of \(-1\).
3. Recall that an even root of a negative number is not a real number.
4. Understand that the sixth root of \(-1\) involves finding a number that, when raised to the sixth power, equals \(-1\).
5. Conclude that since no real number raised to an even power can be negative, \( \sqrt[6]{-1} \) is not a real number.

Key Concepts

Roots of Numbers

Roots are mathematical operations that determine a number which, when raised to a specific power, yields the original number. For example, the square root of 9 is 3, since 3² = 9. In this case, we are dealing with a sixth root, which means we are looking for a number that, when raised to the sixth power, equals -1.

Real and Complex Numbers

Real numbers include all the numbers on the number line, encompassing both positive and negative values, as well as zero. However, certain roots, such as the sixth root of -1, do not yield real numbers. Instead, they fall into the category of complex numbers, which include imaginary units represented as 'i', where i² = -1.

Even Roots of Negative Numbers

When taking even roots (like square roots, fourth roots, etc.) of negative numbers, the result is not a real number. This is because no real number squared or raised to an even power can produce a negative result. Therefore, the sixth root of -1 is not a real number, but can be expressed in terms of complex numbers.