Question 65

The sum of all rational terms in the expansion of $$\left(2^{\frac{1}{5}} + 5^{\frac{1}{3}}\right)^{15}$$ is equal to:

General term: T(r+1) = C(15,r) × 2^((15-r)/5) × 5^(r/3)

For rational terms: (15-r)/5 and r/3 must be non-negative integers.

15-r must be divisible by 5: r = 0, 5, 10, 15

r must be divisible by 3: r = 0, 3, 6, 9, 12, 15

Common values: r = 0 and r = 15.

T(1) = C(15,0) × 2³ × 1 = 8

T(16) = C(15,15) × 1 × 5⁵ = 3125

Sum = 8 + 3125 = 3133

The correct answer is Option 1: 3133.

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