<p>Express the following complex numbers in \(a + ib\) form: (b) \(\dfrac{2 - \sqrt{-25}}{1 - \sqrt{-16}}\)</p>
Step-by-Step Solution
Key Concept: Simplify imaginary components using √(-n) = i√n, then rationalize by multiplying by the conjugate of the denominator to eliminate imaginary parts from the denominator.
<p><strong>Step 1:</strong> Convert square roots of negative numbers to imaginary form.</p><p>√(-25) = i√25 = 5i</p><p>√(-16) = i√16 = 4i</p><p>So the expression becomes: <strong>(2 - 5i)/(1 - 4i)</strong></p><p><strong>Step 2:</strong> Multiply numerator and denominator by the conjugate of the denominator.</p><p>Conjugate of (1 - 4i) is (1 + 4i)</p><p><strong>Step 3:</strong> Expand numerator: (2 - 5i)(1 + 4i)</p><p>= 2(1) + 2(4i) - 5i(1) - 5i(4i)</p><p>= 2 + 8i - 5i - 20i²</p><p>= 2 + 3i - 20(-1)</p><p>= 2 + 3i + 20 = <strong>22 + 3i</strong></p><p><strong>Step 4:</strong> Expand denominator: (1 - 4i)(1 + 4i)</p><p>= 1² - (4i)² = 1 - 16i² = 1 - 16(-1) = 1 + 16 = <strong>17</strong></p><p><strong>Step 5:</strong> Divide: (22 + 3i)/17</p><p>∴ Answer: <strong>22/17 + 3i/17</strong> or <strong>(22 + 3i)/17</strong></p>
Correct Answer: 22