Simplify the complex number expression: (3 + 4i) / (1 - 2i), where i = sqrt(-1).
Answer and explanation
Correct answer: B. -1 + 2i
- Step 1: Multiply numerator and denominator by the complex conjugate of the denominator, (1 + 2i):
- Step 2: Numerator: (3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i^2 = 3 + 10i + 8(-1) = -5 + 10i.
- Step 3: Denominator: (1 - 2i)(1 + 2i) = 1 - 4i^2 = 1 - 4(-1) = 1 + 4 = 5.
- Step 4: Divide each part by 5: (-5 + 10i) / 5 = -1 + 2i.
Answer
-1 + 2i.
Why not the others: '1 + 2i' has a sign error on the real part; '-1 - 2i' has a sign error on the imaginary part; '11/5 + 2i' fails to account for i^2 = -1.
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