In this explainer, we will learn how to identify the cubic roots of unity using de Moivreβs theorem.

One of the interesting and useful applications of de Moivreβs theorem is finding the roots of unity, by which we mean, for some , finding all the complex numbers such that . The roots of unity play an important role in group theory, number theory, and discrete Fourier transforms. In this explainer, we will focus on the cubic roots of unity. We will begin by using de Moivreβs theorem to find the three cubic roots of one.

### Example 1: Cubic Roots of Unity

Find all the values of for which .

### Answer

There are, in fact, multiple methods we could use to solve this. We will demonstrate two. The first is by using de Moivreβs theorem for roots which states that, for a complex number , the roots are given by for . We begin by expressing 1 in polar form as

Therefore, the cubic roots of one are given by

for . Starting with , we get . For , we have

Finally, for , we have

Since the argument of this complex number is not in the range , we can subtract so that we can express it with its principal argument as

Therefore, the cubic roots of one are

We can express these in algebraic form as

An alternative way to find the cubic roots of one is by solving the equation . By inspection, we can see that 1 is a trivial solution to this equation. We can, therefore, factor out from this equation and solve the resulting quadratic. Firstly, to factor from the equation, we can write

Expanding the right-hand side, we have

Equating coefficients, we immediately see that and ; using this, we quickly see that too.

Hence,

Using the quadratic formula or otherwise, we can solve to find

Hence, the three cubic roots of one are

If we compare these with the answers we got using de Moivreβs theorem, we see that they are the same.

We will now consider the product of the cubic roots of unity.

### Example 2: Products of the Cubic Roots of Unity

Let and be the complex cubic roots of unity.

- Evaluate . How does this compare with ?
- Evaluate . How does this compare with ?

### Answer

**Part 1**

Using the rules of integer exponents, we can write

Subtracting from the argument, we can express this in exponential form with its principal argument as

Hence, we find that .

**Part 2**

Similarly, using the rules of exponents, we can write

Adding from the argument, we can express this in exponential form with its principal argument as

Hence, we find that .

The results from the previous example can be extended to higher powers of . For example, if we are given , we can express as , where is 0, 1, or 2. Hence, we can write

By definition, . Hence, reduces to , where . Using the result of the previous example, we can summarize this as follows: for positive ,

### Definition: Cubic Roots of Unity

There are three cubic roots of unity. We often use the symbol to represent the primitive root which is the root with the smallest strictly positive argument. We can express in exponential, polar, and algebraic form as follows:

Sometimes is somewhat ambiguously referred to as βtheβ cubic root of unity or βtheβ complex cubic root of unity. We can represent all three cubic roots of unity as 1, , and .

As we have seen, the cubic roots of unity form a cycle under multiplication as represented in the figure below.

In the next question, we will explore the properties of the cubic root of unity when raised to negative powers.

### Example 3: Cubic Roots of Unity and Negative Powers

Let be the primitive cubic root of unity.

- Find . How is this related to the other cubic roots of unity?
- Find . How is this related to the other cubic roots of unity?

### Answer

**Part 1**

Expressing in exponential form and using the properties of integer exponents, we have

Clearly this is equal to .

**Part 2**

Similarly, using the exponential form, we have

Since, in this form, the argument is not between and , we can add to represent it in exponential form with the principal argument as which we can see is equal to .

The previous example demonstrates that the cubic roots of unity also form a cycle under division.

We can summarize the properties of the cubic roots of unity under multiplication and division as for any .

Plotting the cube roots of unity on an Argand diagram highlights their symmetry. Furthermore, we observe that

Noticing this will help us as we consider the properties of the cubic roots of unity under addition and subtraction.

### Example 4: Sums and Differences of the Cubic Roots of Unity

Let be the primitive cubic root of unity.

- Find .
- Find .
- What is and how is it related to the other roots of unity?
- What is and how is it related to the other roots of unity?

### Answer

**Part 1**

Given that , we can rewrite this expression as

From the properties of complex conjugates, we have

Using the algebraic form , we see

An alternative way we could have derived this result is by noting that . Hence,

Since , we know that . Hence,

**Part 2**

Once again, using the fact that , we can rewrite the expression as

Using the properties of conjugation, we have

Considering the algebraic form , we see

**Part 3**

Using the algebraic form of , we can write

To see how this is related to the cubic roots of unity, we first notice that both and have negative real parts. Hence, if we rewrite this as we can see that . We could have also derived this by rearranging the formula ; this is the approach we will take to answer the next part of the question.

**Part 4**

Given the fact that , by subtracting from both sides, we get

Hence,

The previous example demonstrated a number of properties of the cubic roots of unity which we summarize below.

### Properties of the Cubic Roots of Unity

The cubic roots of unity have the following properties:

- ,
- ,
- .

We will now consider some examples which will demonstrate how we can use the properties of the cubic roots of unity to solve problems.

### Example 5: Using the Properties of the Cubic Roots of Unity

Evaluate where is a complex cube root of unity.

### Answer

Using the properties of the cubic roots of unity, we know that

Hence,

Using the properties of integer indices, we have

### Example 6: Simplifying Expressions Using the Properties of the Cubic Roots of Unity

Evaluate where is a complex cube root of unity.

### Answer

Let us consider some of the properties of the cubic roots of unity that might help us simplify this expression. We notice that in the denominator of the fraction in the first set of parenthesis is the term . From the properties of the cubic roots of unity, we know that . We also notice that, in the second set of parenthesis, we have , which we can also simplify using the same result. Hence, we can rewrite the expression as

We can now use the properties of the powers of the cubic roots of unity to replace with and with .

Hence,

Since , we have

### Example 7: Simplifying Expressions Using the Properties of the Cubic Roots of Unity

Evaluate .

### Answer

We begin by replacing higher powers of with their equivalent power between 0 and 2. Hence, replacing with , we have

Factorizing out the constants in each set of parenthesis gives

Using the properties of the cubic roots of unity, we can rewrite this as

Since , finally, we can state that

Many of the properties of the cubic roots of unity have their analogues in the more general case of the roots of unity.

### Key Points

- There are three cubic roots of unity which we denote 1, , and , where is referred to as the primitive cubic root of unity. We can express it in exponential, polar, and algebraic forms as
- Both the positive and the negative powers of form a closed cycle of three elements, which we can summarize as for any .
- The cubic roots of unity also have the following properties:
- ,
- ,
- .

- We can use the properties of the cubic roots of unity to simplify complex looking expressions.