A large power can become simple when the base satisfies a short cycle. For roots of unity, combine that cycle with a relationship between the roots rather than expanding repeated complex products.
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Identify the three roots
The roots of z3 = 1 are 1, ω and ω2, where ω is a nonreal cube root. They satisfy ω3 = 1 and 1 + ω + ω2 = 0.
The second relationship follows from z3 - 1 = (z - 1)(z2 + z + 1), evaluated at ω ≠ 1.
Reduce the exponent
Since 2026 leaves remainder 1 on division by 3, the 2026th powers of the three roots are 1, ω and ω2. Their sum is zero.
By contrast, 2025 is divisible by three, so every root raised to that power equals 1. The sum is then 3.
Distinguish one root from the complete set
For an unspecified nonreal root, its 2026th power equals that same root; the information does not choose between the two nonreal values.
A symmetric sum over all roots can be determined even when an individually labelled value is not fixed.
Check the polynomial before generalising
This short cycle depends on the equation z3 = 1. Roots of another polynomial may have a different structure.
This is original algebra practice for AMC 12 preparation, not an official paper prediction. Derive the root relationship once, then explain precisely which remainder rule you are using.
Continue reading: AMC 12 Trigonometry: Solve Equations Without Losing Solutions.
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Official information and further reading: Source 1
