The sum of all rational terms in the expansion of (2^(1/5)+5^(1/3))¹⁵ is equal to :

Solution

Maths sir · 4 steps

Write the general term and ask when both powers become whole numbers. Only the two end terms pass.

1. General term. The (r+1)th term is Tᵣ₊₁ = C(15, r) 2^((15-r)/5) 5^(r/3), with r = 0, 1, …, 15.
2. Rational condition. The term is rational only when both exponents are integers. So r must be a multiple of 3, and 15 - r must be a multiple of 5, which means r is also a multiple of 5.

   r (multiple of 3)	(15-r)/5	Rational?
   0	3	yes
   3	12/5	no
   6	9/5	no
   9	6/5	no
   12	3/5	no
   15	0	yes

3. Add the terms. For r = 0: C(15, 0) 2³ = 8. For r = 15: C(15, 15) 5⁵ = 3125.
4. So. The sum of the rational terms is 8 + 3125 = 3133, which is option (c).

The idea to keep. For rational terms, make every exponent in Tᵣ₊₁ an integer. Here r must be a multiple of both 3 and 5, so only multiples of LCM(3,5) = 15 survive.

Answer from the NTA final answer key. Solution by PrepWiser.