<p>If <span>\(z\)</span> is a nonreal root of <span>\(\sqrt[7]{-1}\)</span>, then find the value of <span>\(z^{86} + z^{175} + z^{289}\)</span>.</p>
Step-by-Step Solution
Key Concept: The 7th roots of -1 are of the form z = e^(iπ(2k+1)/7) for k = 0,1,...,6, and since z^7 = -1, we can reduce any power of z modulo 7 using z^7 = -1 to simplify the expression.
<p><strong>Step 1:</strong> Since z is a nonreal 7th root of -1, we have z^7 = -1.</p><p><strong>Step 2:</strong> Reduce each exponent modulo 7 using z^7 = -1:<br/>• 86 = 7(12) + 2, so z^86 = (z^7)^12 · z^2 = (-1)^12 · z^2 = z^2<br/>• 175 = 7(25) + 0, so z^175 = (z^7)^25 = (-1)^25 = -1<br/>• 289 = 7(41) + 2, so z^289 = (z^7)^41 · z^2 = (-1)^41 · z^2 = -z^2</p><p><strong>Step 3:</strong> Compute the sum:<br/>z^86 + z^175 + z^289 = z^2 + (-1) + (-z^2) = z^2 - 1 - z^2 = -1</p><p>∴ Answer: <strong>-1</strong></p>
Correct Answer: -1