<p>Let \(z\) be a complex number satisfying the equation \((z^3 + 3)^2 = -16\), then find the value of \(|z|\).</p>
Step-by-Step Solution
Key Concept: Taking square roots of both sides gives z³ + 3 = ±4i, then recognizing that |z³ + 3| = 4 allows you to find |z³| directly using the relationship |z³ + 3|² = |z³|² + 2Re(3z̄³) + 9, or more elegantly: if z³ = w, then |w + 3| = 4 means w lies on a circle, and the key is that |z| depends only on the magnitude constraint.
<p><strong>Step 1:</strong> Take modulus of both sides of (z³ + 3)² = -16.</p><p>|(z³ + 3)²| = |-16|</p><p>|z³ + 3|² = 16</p><p>|z³ + 3| = 4</p><p><strong>Step 2:</strong> Let w = z³, so |w + 3| = 4. This means w lies on a circle of radius 4 centered at -3.</p><p><strong>Step 3:</strong> The modulus |w| is minimized when w lies on the real axis between -3 and the origin. From |w + 3| = 4 with w real and negative:</p><p>w + 3 = ±4</p><p>w = 1 or w = -7</p><p><strong>Step 4:</strong> Since (z³ + 3)² = -16 (purely imaginary), we need z³ + 3 = ±4i. Therefore:</p><p>|z³| must satisfy the constraint. Testing: if z³ = -3 ± 4i, then |z³|² = 9 + 16 = 25</p><p>So |z³| = 5, giving |z|³ = 5</p><p><strong>Step 5:</strong> Therefore, |z| = 5^(1/3)</p><p>∴ Answer: <strong>5^(1/3)</strong></p>
Correct Answer: 5^(1/3)