ACT Math Practice Question #871 (Hard (36)) | Test Citadel
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ACT Math Difficulty: Hard (36)

Digital ACT Math Practice Question #871

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For the imaginary unit \( i = \sqrt{-1} \), which of the following expressions is equivalent to: \[ \frac{5 + 2i}{3 - 4i} \]?
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Socratic AI Engine Step-by-Step Derivation
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Tactical Insight (Hint 1)

To eliminate \( i \) from the denominator, multiply both numerator and denominator by the complex conjugate of \( 3 - 4i \), which is \( 3 + 4i \).

Elimination Framework (Hint 2)

Remember that \( i^2 = -1 \). In the numerator, \( 8i^2 \) becomes \( 8(-1) = -8 \), which subtracts from 15 to give 7.

Masterclass Solution & Distractor Trap Analysis

To simplify a complex quotient, multiply the numerator and denominator by the complex conjugate of the denominator \( (3 + 4i) \): \[ \frac{(5 + 2i)(3 + 4i)}{(3 - 4i)(3 + 4i)} \] Denominator: \( 3^2 - (4i)^2 = 9 - 16(-1) = 9 + 16 = 25 \). N...

Distractor Analysis: Trap choice eliminates careless test-takers who confuse roots with coordinates...

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