If x = a + b, y = aγ + bβ and z = aβ + bγ where γ and β are the complex cube roots of unity, show th

Q: Wait, does it matter if I assign $\gamma = \omega^2$ and $\beta = \omega$ instead?

Not at all! The expressions for $y$ and $z$ are symmetric with respect to $\omega$ and $\omega^2$. Swapping them simply swaps $y$ and $z$, but their product $yz$ remains invariant. This is the beauty of symmetric polynomials.

Q: Why does $\omega^4$ simplify to $\omega$?

Remember that $\omega^3 = 1$. Therefore, $\omega^4 = \omega^3 \cdot \omega = 1 \cdot \omega = \omega$. It is a cyclic property—every power of $\omega$ greater than 2 can be reduced to either $1, \omega,$ or $\omega^2$.

Q: Can I solve this using determinants?

Yes, you absolutely can! This expression is actually the determinant of a circulant matrix. While that is a more advanced perspective, the algebraic expansion method we used is the most direct and intuitive way to see the identity emerge.

Question

If $x = a + b, y = a\gamma + b\beta$ and $z = a\beta + b\gamma$ where $\gamma$ and $\beta$ are the complex cube roots of unity, show that $xyz = a^3 + b^3$.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are given the variables:
$x = a + b$
$y = a\omega + b\omega^2$
$z = a\omega^2 + b\omega$

Here, $\omega$ and $\omega^2$ represent the complex cube roots of unity. These roots satisfy the fundamental properties:
$$\omega^3 = 1$$
$$1 + \omega + \omega^2 = 0$$

From the second property, we derive the identity $\omega + \omega^2 = -1$, which will serve as our primary tool for simplification.

### The Master Equation

To find the product $xyz$, we substitute the given expressions:
$$xyz = (a + b)(a\omega + b\omega^2)(a\omega^2 + b\omega)$$

We keep the first term $(a + b)$ constant and focus on expanding the product of the latter two brackets:
$$(a\omega + b\omega^2)(a\omega^2 + b\omega)$$

Expanding this product term by term yields:
$$a^2\omega^3 + ab\omega^2 + ab\omega^4 + b^2\omega^3$$

### Simplifying the Expression

Using the property $\omega^3 = 1$, we simplify the expression. Note that $\omega^4 = \omega^3 \cdot \omega = \omega$:
$$a^2(1) + ab\omega^2 + ab\omega + b^2(1)$$
$$= a^2 + ab(\omega^2 + \omega) + b^2$$

Substituting the identity $\omega + \omega^2 = -1$ into the middle term:
$$= a^2 + ab(-1) + b^2$$
$$= a^2 - ab + b^2$$

### Final Calculation

Now, we multiply this result by the first bracket $(a + b)$ that we set aside earlier:
$$xyz = (a + b)(a^2 - ab + b^2)$$

Recognizing this as the standard algebraic identity for the sum of two cubes, we conclude:
**$xyz = a^3 + b^3$**

Animated Solution for Mathematics - Complex Numbers: If $x = a + b, y = aγ + b β$ and $z = a β + bγ$ where $γ$ and $β$ are the complex cube roots of unity, show that $x y z = a^{3} + b^{3}$ .

Visualized Solution

Identifying the Roots $γ$ and $β$

Setting up the Product $x y z$

Expanding the Complex Terms

Simplifying Powers of $ω$

Applying the Sum Property

The Final Identity $a^{3} + b^{3}$

The Sigma Insight: Cube Roots and nth Roots of Unity

Analyzing the Setup

The Master Equation

Simplifying the Expression

Final Calculation

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Animated Solution for Mathematics - Complex Numbers: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x=a+b,y=aγ+bβ and z=aβ+bγ where γ and β are the complex cube roots of unity, show that xyz=a3+b3.

Visualized Solution

Identifying the Roots .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } γ and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } β

Setting up the Product .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } xyz

Expanding the Complex Terms

Simplifying Powers of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ω

Applying the Sum Property

The Final Identity .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a3+b3

The Sigma Insight: Cube Roots and nth Roots of Unity

Analyzing the Setup

The Master Equation

Simplifying the Expression

Final Calculation

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