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Sum of all rational terms in an expansion with surds

This type of question, asked 4 times: 4 Apr 2024, 23 Jan 2025, 28 Jan 2025, 3 Apr 2025. All 4 questions below, oldest first.

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  1. Question 1

    4 Apr 2024

    70
    The sum of all rational terms in the expansion of (215+513)15\left( 2^{\frac{1}{5}}+5^{\frac{1}{3}} \right)^{15} is equal to:

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  2. Question 2

    23 Jan 2025

    72
    The sum of all rational terms in the expansion of (1+21/3+31/2)6\left( 1 +2^{1/3}+3^{1/2} \right)^{6} is equal to ____\_\_\_\_

  3. Question 3

    28 Jan 2025

    143
    Let the coefficients of three consecutive terms TrT_{r}, Tr+1T_{r + 1} and Tr+2T_{r + 2} in the binomial expansion of (a+b)12(a + b)^{12} be in a G.P. and let p be the number of all possible values of rr. Let qq be the sum of all rational terms in the binomial expansion of (34+43)12(\sqrt[4]{3}+\sqrt[3]{4})^{12}. Then p+qp + q is equal to :

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  4. Question 4

    3 Apr 2025

    54
    The sum of all rational terms in the expansion of (2+3)8(2 +\sqrt{3})^{8} is

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