 In general, we can find the nth roots of a complex number z by writing z in trigonometric form as r-sys-theta plus some multiple of 2π, and then computing nth root of r-sys-theta-nths plus 2πk-nths. So for example, let's find all cube roots of 1 plus i. Rewriting 1 plus i in trigonometric form gives us... So any cube root must satisfy. And so r-cubed is square root of 2, and so r is the 6th root of 2, and 3-theta must be π-4th plus 2πk, and so theta will be... with distinct values for k equals 0, 1, and 2, giving us our arguments. And so the cube roots will be... And remember we should answer questions in the same language they were given, so here we have our complex number in rectangular form, which means we should write our answer in rectangular form. And that means we need to find cosine of π-12 and sine of π-12. Fortunately, we remember those half-angle formulas, and since π-12 is a first quadrant angle, we'll use the positive square root and find. For the others, remember the arguments of the cube third root differ by 2π-3, and we have our angle sum and difference formulas. So cosine of 9π-12, well, that's really the cosine of π-12 plus 2π-3, and sine of 9π-12 is sine of π-12 plus 2π-3, and we find... And this will be... Well, while we could do this, all we're really doing here is reviewing trigonometric identities, and we're not really doing complex analysis. Even though we did get the question in rectangular form, and we really should write our answer in rectangular form, we'll keep in mind that a foolish consistency is the hobgoblin of little minds, and we'll leave our answers in trigonometric form. And that's because we really want to focus on the complex analysis. We're now in a position to define the principal root of a complex number. So let's try the following. Let z equal r sis theta with r a non-negative real number and theta the least non-negative argument of z. Then the principal root will be... There's just one problem with this definition. This doesn't work. To see why, let's try to find all cube roots of negative 1 and identify the principal cube root. We find that negative 1 is sis π, so our cube roots will be... Since π is the least argument of z, our definition would make the principal root this one. But we generally define the principal cube root of negative 1 to be negative 1. So which is it? It's best to avoid too many inconsistencies in mathematics. Sorry Ralph. And we like the nth root of a to be a real number whenever possible. And so we define the principal root of a complex number as follows. The principal nth root of z is... The non-negative number whose nth power is z. Or if no such number exists, the negative real number whose nth power is z. Or if no such number exists, the nth root of r sis theta n squared z equals r sis theta with r a non-negative real number and theta the least non-negative argument of z. Phew. The thing is, mathematicians typically don't like complicated definitions like this. And as a result, we don't often ask about the principal nth root of a complex number. Instead, we're more interested in what's known as a primitive nth root. And we'll take a look at that later.