Question

Every nonzero complex number has exactly _______ distinct complex cube roots.

Short answer

Three distinct complex cube roots.

Step-by-step solution

Understand Cube Roots

Recall that a cube root of a complex number is a number that, when raised to the power of 3, yields the original number. For any nonzero complex number, there are multiple roots.

Use De Moivre's Theorem

De Moivre's Theorem helps in finding the cube roots of a complex number. If a complex number is written in polar form as \[ z = r e^{i \theta} \], its cube roots can be derived using the formula \[ z^{1/3} = r^{1/3} e^{i (\theta + 2k\pi)/3} \] for \ k = 0, 1, 2 \.

Identify Distinct Roots

Notice that the cube root formula \[ z^{1/3} = r^{1/3} e^{i (\theta + 2k\pi)/3} \] yields different values based on the integer \ k \. Since \ k \ takes values 0, 1, and 2, it generates three distinct cube roots.

Summarize the Result

Since we identified three different cube roots for each nonzero complex number, these are exactly all the possible distinct cube roots.

Key concepts

complex numbers

A complex number combines a real part and an imaginary part. It has the general form \(z = a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The imaginary unit \(i\) is defined by the property \(i^2 = -1\). This might sound abstract, but it allows us to solve equations that don't have solutions within the real numbers alone.
Complex numbers are often visualized on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. This visual aid makes it easier to grasp their behavior and operations.

cube roots

To understand cube roots of complex numbers, we need to think about roots in general. A cube root of a number \(z\) is a number that, when cubed, gives \(z\). For complex numbers, this process involves not just finding the magnitude (distance from the origin in the complex plane) but also the angle (or argument) with respect to the positive real axis.
For every nonzero complex number, there are exactly three distinct cube roots. These roots are reached by solving the equation \[ w^3 = z \]. Using polar form and De Moivre's Theorem simplifies this process.

De Moivre's Theorem

De Moivre's Theorem is a powerful tool in complex number theory. It connects trigonometry and complex numbers by stating that \[(re^{i\theta})^n = r^n e^{i n\theta} \].
This theorem allows for easy computation of powers and roots of complex numbers when they are in polar form. In finding cube roots, De Moivre’s Theorem simplifies the process by letting us adjust the angle divided by three while considering all possible rotations around the circle.
The general formula for cube roots of a complex number using De Moivre's Theorem is:
\ z^{1/3} = r^{1/3} e^{i (\theta + 2k\frac{ \pi}3)} \ , \ k = 0, 1, 2 \.

polar form

Complex numbers can be represented in polar form, which is particularly useful for multiplication, division, and finding roots. The polar form of a complex number \(z\) is \(z = r e^{i \theta}\), where \(r\) is the magnitude (or modulus) and \(\theta\) is the argument (or angle).
Converting to polar form involves finding \(r\) using \[ r = \sqrt{a^2 + b^2} \]
and \(\theta\) using \[ \theta = \tan^{-1} \frac{b}{a} \].
This form reveals more about the geometric interpretation of complex numbers and makes operations more straightforward. Using it with De Moivre's Theorem lets us find cube roots conveniently by manipulating \(r\) and \(\theta\).