<p>If <span>\(\sqrt{x+iy} = \pm(a+ib)\)</span>, then find <span>\(\sqrt{-x-iy}\)</span>.</p>
Step-by-Step Solution
Key Concept: If √(x+iy) = ±(a+ib), then squaring gives x+iy = (a+ib)². Use the relationship between x+iy and -x-iy by recognizing that -x-iy = -1·(x+iy), so √(-x-iy) = √(-1)·√(x+iy) = i·(±(a+ib)).
<p><strong>Step 1:</strong> Given √(x+iy) = ±(a+ib), square both sides to verify: (a+ib)² = a² - b² + 2abi = x + iy</p><p><strong>Step 2:</strong> Recognize that -x-iy = -(x+iy). Therefore:</p><p>√(-x-iy) = √(-(x+iy)) = √(-1)·√(x+iy) = i·(±(a+ib))</p><p><strong>Step 3:</strong> Compute i(a+ib) = ia + i²b = ia - b = -(b - ia)</p><p><strong>Step 4:</strong> Since we have ±, we get √(-x-iy) = ±i(a+ib) = ±(ia-b) = ∓(b-ia) = ±(b-ia)</p><p>∴ Answer: <strong>±(b - ia)</strong></p>
Correct Answer: ±(b - ia)