<p>Find the multiplicative inverse of the complex number \(4 - 3i\).</p>
Step-by-Step Solution
Key Concept: The multiplicative inverse of a complex number z is found by dividing 1 by z, which requires multiplying numerator and denominator by the complex conjugate to eliminate the imaginary part from the denominator.
<p><strong>Step 1:</strong> For complex number z = 4 - 3i, the multiplicative inverse is z⁻¹ = 1/z = 1/(4-3i)</p><p><strong>Step 2:</strong> Multiply numerator and denominator by the complex conjugate of the denominator (4+3i):</p><p>z⁻¹ = 1/(4-3i) × (4+3i)/(4+3i) = (4+3i)/[(4-3i)(4+3i)]</p><p><strong>Step 3:</strong> Compute the denominator using (a-b)(a+b) = a² - b²:</p><p>(4-3i)(4+3i) = 4² - (3i)² = 16 - 9i² = 16 - 9(-1) = 16 + 9 = 25</p><p><strong>Step 4:</strong> Simplify:</p><p>z⁻¹ = (4+3i)/25 = 4/25 + 3i/25</p><p>∴ Answer: <strong>4/25 + 3i/25</strong> or <strong>(4+3i)/25</strong></p>
Correct Answer: 4