Final answer:
To solve the equation z³ + 1 = -i, one needs to express it in a way that leverages the sum of cubes and then convert to polar form to find the roots. The provided reference does not contain the necessary steps to reach the actual solution.
Step-by-step explanation:
The student is asked to solve the equation z³ + 1 = -i. This equation cannot be directly solved using common algebraic methods, but it resembles the equation for the sum of cubes, a³ + b³ = (a + b)(a² - ab + b²), where we can assign a = z and b = 1. Since -i is on the right side, we need to express 1 in terms of imaginary units as well where 1 = 0+i. Then our equation can be rewritten in terms of sum of cubes and factored using complex numbers.
The first step in solving this equation involves recognizing that ³√-i needs to be expressed in polar form. The standard rectangular form of a complex number is a + bi, which can be converted to polar form, r(cosθ + i sinθ), using the conversion formulas r = √(a² + b²) and θ = atan2(b,a). From this we can obtain the roots of the equation. However, the actual solution steps for this equation are not provided in the reference information given.