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Home » The Shocking Truth: How Many Water Bottles Truly Fill a Gallon—Facts You Need! - Londonproperty

The Shocking Truth: How Many Water Bottles Truly Fill a Gallon—Facts You Need! - Londonproperty

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 z^6 + z^3 + 1 = 0.

📌 Let $ w = z^3 $. Then:

📌 So $ z^3 = \omega $ or $ z^3 = \omega^2 $, where $ \omega = e^{2\pi i / 3} = \frac{-1 + i\sqrt{3}}{2} $, a primitive cube root of unity. The other root is $ \omega^2 = e^{-2\pi i / 3} $.

📌 We now solve $ z^3 = \omega $ and $ z^3 = \omega^2 $. Each has three cube roots.

📌 First, write $ \omega = e^{2\pi i / 3} $, so the cube roots are:

📌 z = \omega^{1/3} = e^{2\pi i / 9},\ e^{2\pi i (1+3)/9} = e^{8\pi i / 9},\ e^{2\pi i (1+6)/9} = e^{14\pi i / 9} \quad \text{(mod } 2\pi).

📌 Similarly, $ z^3 = \omega^2 = e^{-2\pi i / 3} $, so roots:

📌 z = e^{-2\pi i / 9},\ e^{-8\pi i / 9},\ e^{-14\pi i / 9} = e^{4\pi i / 9} \quad \text{(since } -14\pi/9 \equiv 4\pi/9 \mod 2\pi).

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