Write each root using exponents and evaluate. ∜-81

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.