<p>Simplify <span>\(\dfrac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}-\sqrt{5-12i}}\)</span>.</p>
Step-by-Step Solution
Key Concept: Convert 5±12i to standard form (a+bi)² to find their square roots, then use the conjugate pair property to simplify the ratio algebraically.
<p><strong>Step 1:</strong> Find √(5+12i). Let √(5+12i) = a+bi. Then (a+bi)² = 5+12i, so a²-b² = 5 and 2ab = 12.</p><p>From 2ab = 12: ab = 6, so b = 6/a. Substituting: a² - 36/a² = 5 → a⁴ - 5a² - 36 = 0 → (a²-9)(a²+4) = 0.</p><p>Thus a² = 9, so a = 3 (taking positive). Then b = 2, giving √(5+12i) = 3+2i.</p><p><strong>Step 2:</strong> Similarly, √(5-12i) = 3-2i (conjugate, since 5-12i is the conjugate of 5+12i).</p><p><strong>Step 3:</strong> Let u = 3+2i and v = 3-2i. The expression becomes:</p><p>$$\frac{u+v}{u-v} = \frac{(3+2i)+(3-2i)}{(3+2i)-(3-2i)} = \frac{6}{4i} = \frac{3}{2i}$$</p><p><strong>Step 4:</strong> Rationalize: $$\frac{3}{2i} \cdot \frac{-i}{-i} = \frac{-3i}{-2i²} = \frac{-3i}{2}$$</p><p>∴ Answer: <strong>-3i/2</strong></p>
Correct Answer: -3i/2