Understanding Cube Roots of Unity

This question asks us to evaluate an expression involving powers of ω, where ω is a cube root of unity. Cube roots of unity are complex numbers that, when cubed, result in 1. The three cube roots of unity are $1$, $\omega$, and $\omega^2$. These roots have important properties that help us simplify expressions.

Key Properties of Cube Roots of Unity

The crucial properties of ω (a non-real cube root of unity) are:

• $\omega^3 = 1$
• $1 + \omega + \omega^2 = 0$
• Higher powers of ω repeat in a cycle of 3: $\omega^n = \omega^{n \pmod 3}$

Evaluating the Expression ω¹⁰⁰ + ω²⁰⁰ + ω³⁰⁰

We need to find the value of $\omega^{100} + \omega^{200} + \omega^{300}$. Let's evaluate each term separately using the property $\omega^n = \omega^{n \pmod 3}$. This means we divide the exponent by 3 and look at the remainder.

Term 1: $\omega^{100}$

Divide the exponent 100 by 3:

$100 \div 3 = 33$ with a remainder of $1$.

So, $100 = 3 \times 33 + 1$.

Using the property $\omega^n = \omega^{n \pmod 3}$ and $\omega^3 = 1$:

$\omega^{100} = \omega^{3 \times 33 + 1} = (\omega^3)^{33} \times \omega^1 = 1^{33} \times \omega = 1 \times \omega = \omega$.

Term 2: $\omega^{200}$

Divide the exponent 200 by 3:

$200 \div 3 = 66$ with a remainder of $2$.

So, $200 = 3 \times 66 + 2$.

Using the property $\omega^n = \omega^{n \pmod 3}$ and $\omega^3 = 1$:

$\omega^{200} = \omega^{3 \times 66 + 2} = (\omega^3)^{66} \times \omega^2 = 1^{66} \times \omega^2 = 1 \times \omega^2 = \omega^2$.

Term 3: $\omega^{300}$

Divide the exponent 300 by 3:

$300 \div 3 = 100$ with a remainder of $0$.

So, $300 = 3 \times 100 + 0$.

Using the property $\omega^n = \omega^{n \pmod 3}$ and $\omega^3 = 1$:

$\omega^{300} = \omega^{3 \times 100 + 0} = (\omega^3)^{100} \times \omega^0 = 1^{100} \times 1 = 1 \times 1 = 1$.

Summing the Terms

Now, substitute the simplified terms back into the original expression:

$\omega^{100} + \omega^{200} + \omega^{300} = \omega + \omega^2 + 1$.

Using the fundamental property of cube roots of unity, $1 + \omega + \omega^2 = 0$.

Therefore, $\omega + \omega^2 + 1 = 0$.

The value of the expression $\omega^{100} + \omega^{200} + \omega^{300}$ is 0.

Summary of Power Simplification

Revision Table: Cube Roots of Unity

Additional Information: Complex Numbers and Roots of Unity

Cube roots of unity are specific examples of roots of unity, which are complex numbers that are solutions to the equation $z^n = 1$ for any positive integer $n$. For $n=3$, the equation is $z^3 = 1$. The solutions can be found using polar form of complex numbers.

• The principal root is $z=1$.
• The other two roots are complex: $\omega = e^{i 2\pi/3} = \cos(2\pi/3) + i \sin(2\pi/3) = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$
• And $\omega^2 = e^{i 4\pi/3} = \cos(4\pi/3) + i \sin(4\pi/3) = -\frac{1}{2} - i\frac{\sqrt{3}}{2}$

The geometric representation of the cube roots of unity is points equally spaced on the unit circle in the complex plane, forming vertices of an equilateral triangle.

Understanding roots of unity is fundamental in areas like abstract algebra, number theory, and digital signal processing.