Complex Numbers
Operations on Complex Numbers
Grade 11
Question:
<p>Find the imaginary part of <i>(3 + 2√(-54))</i><sup>1/2</sup> - <i>(3 - 2√(-54))</i><sup>1/2</sup>.</p>
Step-by-Step Solution
Key Concept: Recognize the radicand as a perfect square of a complex number to simplify the square root computation.
<p><strong>Step 1:</strong> Rewrite using <i>i</i>:</p><p><i>(3 + 2 · 3√6 · i)</i><sup>1/2</sup> - <i>(3 - 2 · 3√6 · i)</i><sup>1/2</sup></p><p><strong>Step 2:</strong> Recognize perfect squares:</p><p>= <i>[(3 + √6 · i)</i><sup>2</sup>]<sup>1/2</sup> - <i>[(3 - √6 · i)</i><sup>2</sup>]<sup>1/2</sup></p><p><strong>Step 3:</strong> Take square roots:</p><p>= ±<i>(3 + √6 · i)</i> ∓ <i>(3 - √6 · i)</i></p><p><strong>Step 4:</strong> Simplify:</p><p>= ±2√6 · i</p><p>∴ Imaginary part = ±2√6</p>
Correct Answer: ±2√6