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📅 March 13, 2026 👤 scraface

Mar 13, 2026

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#### 78.5Question: Let $ a $ and $ b $ be complex numbers such that $...
Solution: We are given that $ a^2 + b^2 = 1 $ and $ ab = \frac{1}{2} $...
We start by using the identity:
a^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2
We already know $ a^2 + b^2 = 1 $ and $ ab = \frac{1}{2} $, so
a^2b^2 = (ab)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}
Substituting into the identity:
a^4 + b^4 = 1^2 - 2 \cdot \frac{1}{4} = 1 - \frac{1}{2} = \frac{1}{2}
|a^4 + b^4| = \left| \frac{1}{2} \right| = \boxed{\frac{1}{2}}
Solution: To minimize

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