How do I find the cube root of a complex number?
To find a cubic root (or generally root of degree
From tis formula you can see, that every complex number has
So to calculate root of a complex number you first have to write the number in a trigonometric form
There are three cube roots:
This is an important question, or is at least hinting at one. Unfortunately I can only get to the question in the usual space allotted. The answer will have to wait.
I'm never sure what precalculus is. What's left after Algebra I, Algebra II, Geometry and Trigonometry? But let's assume it's a time to get deep about what the math we've been doing really means.
Taking the cube root is easy if we have our complex number in polar coordinates. I'll write the polar form as
(Hopefully they do it this way in precalc; it makes everything easy).
Every non-zero complex number has three cube roots. In general, any non-integer exponent, like
Those are some symbols that's say if you want to take the cube root of a complex number, take the (real) cube root of its magnitude, and divide the angle by three. That's one cube root. Then the same with the angle
In theory this procedure works if we're given
In practice this is a bit problematic. It's a big shortcut to the answer, but it generally extracts a cost: we end up either with an approximation or a result expressed with trig functions. Neither is ideal.
The deep question we want to address is how to express
This is related to the denesting of radicals, because
We're especially interested in the case where
I'm being warned this is getting long. We've barely gotten to the issue. The most pressing place this comes up is in the cubic formula, which is like the quadratic formula for cubic equations. The simplest form is called the depressed cubic:
We see the cube root of a square root of what is often a negative number, so the cube root of a complex number in rectangular form.
Let's make up a cubic of this form with one root we know, say
Applying the formula,
Here we're supposed to get two but we get the sum of two cube roots of complex numbers. I want to show you how to denest these, which is kinda similar to factoring.
But Socratic says I'm intimidating you all, so I'll stop here and continue this in some future answer. In the meantime, get a pencil and see if you can make this expression equal to two. Other questions to ponder: What are the other two solutions? Each cube root has three values, what are they? Isn't this nine combinations for the sum