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Counting rational, irrational or integral terms in (a^(1/p) + b^(1/q))^n

This type of question, asked 5 times: 2021 Paper 5, 2021 Paper 15, 11 Apr 2023, 30 Jan 2024, 29 Jan 2025. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 5

    72
    If nn is the number of irrational terms in the expansion of (31/4+51/8)60\left( 3^{1/4}+5^{1/8} \right)^{60}, then (n−1)(n- 1) is divisible by:

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  2. Question 2

    2021 Paper 15

    57
    Find the number of irrational terms in (213+314)12\left( 2^{\frac{1}{3}}+3^{\frac{1}{4}} \right)^{12}.

  3. Question 3

    11 Apr 2023

    86
    The number of integral terms in the expansion of (312+514)680\left( 3^{\frac{1}{2}}+5^{\frac{1}{4}} \right)^{680} is equal to

  4. Question 4

    30 Jan 2024

    86
    Number of integral terms in the expansion of {7(12)+11(16)}824\left\{ 7^{\left( \frac{1}{2} \right)}+11^{\left( \frac{1}{6} \right)} \right\}^{824} is equal to

  5. Question 5

    29 Jan 2025

    58
    The least value of n for which the number of integral terms in the Binomial expansion of (73+1112)n(\sqrt[3]{7}+\sqrt[12]{11})^{n} is 183, is :

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