<p>Find nonzero integral solutions of \(|1 - i|^x = 2^x\).</p>
Step-by-Step Solution
Key Concept: Recognize that |1 - i| = √2, so the equation becomes (√2)^x = 2^x, which simplifies to 2^(x/2) = 2^x. For this to hold, exponents must be equal: x/2 = x, yielding only x = 0.
<p><strong>Step 1:</strong> Calculate the modulus of (1 - i).</p><p>|1 - i| = √(1² + (-1)²) = √2</p><p><strong>Step 2:</strong> Substitute into the original equation.</p><p>(√2)^x = 2^x</p><p><strong>Step 3:</strong> Rewrite √2 as 2^(1/2).</p><p>(2^(1/2))^x = 2^x</p><p>2^(x/2) = 2^x</p><p><strong>Step 4:</strong> Since the bases are equal, equate the exponents.</p><p>x/2 = x</p><p>x - x/2 = 0</p><p>x/2 = 0</p><p>x = 0</p><p><strong>Step 5:</strong> Verify the requirement for nonzero integral solutions.</p><p>The only solution is x = 0, which is not nonzero.</p><p>∴ Answer: No solution</p>
Correct Answer: No solution