Write each root using exponents and evaluate. ∜-81

Verified step by step guidance

1. Recognize that the expression ∜-81 represents the fourth root of -81.
2. Express the fourth root using exponents: ∜-81 = (-81)^{1/4}.
3. Note that taking the fourth root of a negative number in the real number system is not possible, as even roots of negative numbers are not real.
4. Consider the complex number system, where the fourth root of a negative number can be found.
5. Express -81 as a complex number: -81 = 81 * (-1), and use the property that (-1)^{1/4} can be expressed in terms of complex numbers.

Key Concepts

Roots and Exponents

Roots are the inverse operations of exponents. The nth root of a number 'a' is a value 'b' such that b^n = a. For example, the square root of 9 is 3 because 3^2 = 9. In this case, the fourth root (∜) indicates we are looking for a number that, when raised to the fourth power, equals -81.

Negative Numbers and Even Roots

When dealing with even roots, such as the square root or fourth root, the result is typically defined only for non-negative numbers in the real number system. This means that the fourth root of -81 does not yield a real number, as no real number raised to an even power can produce a negative result. Instead, we must consider complex numbers to find a solution.

Complex Numbers

Complex numbers extend the real number system to include solutions to equations that do not have real solutions. A complex number is expressed in the form a + bi, where 'a' is the real part and 'bi' is the imaginary part. For the fourth root of -81, we can express it in terms of complex numbers, leading to solutions involving the imaginary unit 'i', where i^2 = -1.