Final answer:
To solve the equation z ⁸ - i = 0, we need to find the eighth roots of i and substitute them into the equation.
Step-by-step explanation:
The given equation is z ⁸ - i = 0. To solve this equation, we need to isolate z. To do this, we can add i to both sides of the equation: z ⁸ = i. Then, we can take the eighth root of both sides to solve for z: z = ⁸√i.
Now, let's find the eighth roots of i. Remember that i is defined as the imaginary unit, where i² = -1. Since we're looking for the eighth roots, we want to find the values of z such that z⁸ = i. In polar form, i = 1(cos(π/2) + isin(π/2)).
Using De Moivre's theorem, we can express the eighth roots of i as z = √[2](1)(cos(π/16 + kπ/4) + isin(π/16 + kπ/4)), where k is an integer from 0 to 7. Substituting the values of k from 0 to 7 will give us the eight solutions for z.