a + b = 9^t + 12^t = (3^t)^2 + 2^(2t)3^t = 2^(4t), 3^t = y
y^2 + 2^(2t)y – 2^(4t) = 0
y = (-2^t ± √(2^(4t) + (4)(2^(4t)))/2, y > 0
3^t = y = 2^(2t)(√5 – 1)/2
(a^2 + b^2)/(ab)
= ((a + b)^2 – 2ab)/(ab)
= (16^t)^2/((9^t)(12^t)) - 2
= 2^(8t)/((2^(2t))(3^(3t))) – 2
= 2^(6t)/ 3^(3t) – 2
= 2^(6t)/(2^(2t)(√5 – 1)/2)^3 – 2
= (2/(√5 – 1))^3 – 2
= ((√5 + 1)/2)^3 – 2
= (5√5 + 15 + 3√5 +1)/8 – 2
= √5
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