# Cube Root of 1 — Value and Cube Roots of Unity

[Algebra](/content/tag/algebra/index.html)

TL;DR

The cube root of 1 is \( \, 1 \; \text{since} \; 1^3 = 1 \). The equation \( x^3 = 1 \) has three cube roots of unity — \( 1, \omega, \text{ and } \omega^2 \) — and this article gives all three values, the derivation, where they appear, and worked examples.

**Quick Answer:**

**Result:** \( \, 1 \; (\text{the real cube root}) \)

**All three roots of \( x^3 = 1 \):** \( 1,\omega=\frac{-1 + i\sqrt{3}}{2},\omega^2=\frac{-1 - i\sqrt{3}}{2} \)

**Notation:** \( 1^{1/3} \; \text{or} \; \sqrt[3]{1} \)

**Method shown:** factoring \( x^3 - 1 = 0 \)

**Exact form:** 1 (real); the other two are complex.

## Quick Reference Table

| Number         | Real cube root | Perfect cube? |
| --------------- | --------------- | -------------- |
| \(\sqrt[3]{1} \) | \( 1 \)       | Yes           |
| \(\sqrt[3]{8} \) | \( 2 \)       | Yes           |
| \(\sqrt[3]{27} \) | \( 3 \)       | Yes           |
| \(\sqrt[3]{64} \) | \( 4 \)       | Yes           |
| \(\sqrt[3]{125} \) | \( 5 \)       | Yes           |
| \(\sqrt[3]{-1} \) | \( -1 \)      | Yes           |

## Where the Cube Roots of 1 Appear

The three cube roots of unity are spaced evenly around a circle of radius 1 in the complex plane, 120° apart. They show up in the discrete Fourier transform used in signal processing, in group theory as the simplest non-trivial cyclic group, and in solving cubic equations by Cardano's method. Anywhere a rotation by a third of a turn matters, \( \omega \) is the number doing the rotating.

## What Is the Cube Root of 1?

The cube root of a number is the value that, multiplied by itself three times, gives that number. Since \( 1 \times 1 \times 1 = 1 \), the real cube root of 1 is 1.

But "the cube root" hides something. The equation \( x^3 = 1 \) is a cubic, and a cubic always has three roots. One is the real number 1; the other two are complex. Together they are called the **cube roots of unity**, where _unity_ just means the number 1. These build on the ideas in [exponents](/content/math/algebra/exponents/index.html) and radicals.

## How to Find the Three Cube Roots of 1 (Methods)

### **Method 1: Factor \( x^3 - 1 = 0 \)**

Start from the definition. Any cube root of 1 satisfies:

\( x^3 = 1 \) → \( x^3 - 1 = 0 \)

Factor the difference of cubes.

\( (x - 1)(x^2 + x + 1) = 0 \)

The first factor gives the real root. \( x - 1 = 0 \) gives \( x = 1 \).

The second factor is a quadratic. Solve \( x^2 + x + 1 = 0 \) with the quadratic formula:
\[ x = \frac{-1 \pm \sqrt{-3}}{2} \]

**Final answer:** the three cube roots of 1 are \( 1, \omega = \frac{-1 + i\sqrt{3}}{2}, \omega^2 = \frac{-1 - i\sqrt{3}}{2} \).

### **Method 2: Use the two key properties**

The complex roots satisfy two relationships:

1. The sum of all three roots is zero: \( 1 + \omega + \omega^2 = 0 \).
2. The product of the three roots is 1: \( 1 \cdot \omega \cdot \omega^2 = 1 \).

These hold because one complex root is the square of the other: squaring \( \omega \) produces \( \omega^2 \), and cubing either returns to 1.

## Common Mistakes With Cube Root of 1

### **Mistake 1: Saying 1 has only one cube root**

**Where it slips in:** when a student meets \( \sqrt[3]{1} \) before complex numbers. **Don't do this:** stopping at \( \sqrt[3]{1} = 1 \) for the equation \( x^3 = 1 \). **The correct way:** the symbol \( \sqrt[3]{1} \) means the principal (real) root, which is 1, but the equation has three solutions.

### **Mistake 2: Dropping the cube-root index**

**Where it slips in:** writing the radical in a hurry. **Don't do this:** writing \( \sqrt{1} \) when you mean the cube root. **The correct way:** show the index: \( \sqrt[3]{1} \).

### **Mistake 3: Forgetting \( \omega^3 = 1 \) when simplifying powers**

**Where it slips in:** evaluating a high power like \( \omega^7 \). **Don't do this:** multiplying \( \omega \) seven times. **The correct way:** reduce the exponent using \( \omega^3 = 1 \), so \( \omega^7 = \omega \).

## Frequently Asked Questions

**What is the cube root of 1?**

The real cube root of 1 is 1, since \( 1^3 = 1 \).

**How many cube roots does 1 have?**

Three. The equation \( x^3 = 1 \) has one real root, 1, and two complex roots, \( \omega \) and \( \omega^2 \).

**What is omega (\( \omega \)) in the cube roots of unity?**

\( \omega \) is one of the two complex cube roots of 1, equal to \( \frac{-1 + i\sqrt{3}}{2} \). Its square is the third root, \( \omega^2 \).

**What is \( 1 + \omega + \omega^2 \)?**

It equals 0. The sum of the three cube roots of unity is always zero.

**Is the cube root of 1 the same as the cube root of −1?**

No. \( \sqrt[3]{1} = 1 \) and \( \sqrt[3]{-1} = -1 \). Both have three roots overall, but their real roots differ in sign.