Final answer:
The middle term in the expansion of (3x/7-2y)¹⁰ is the 6th term, or the 5th term when considering the index starting from 0. It is calculated using the binomial theorem and involves principles like Division of Exponentials. The exact coefficient of the term requires utilizing the combination formula and considering the relevant exponents and signs.
Step-by-step explanation:
The middle term in the expansion of (3x∗2y)¹⁰ is found using the binomial theorem. In the expansion of (a - b)¹⁰, the middle term would occur at the 6th term for a 10-term expansion, which is the 5th index (since the counting starts from 0). The general term in a binomial expansion is given by Tₙ₌₁ = ¹⁰Cₘ a¹⁰−ₘ bₘ, where 'n' is the total number of terms and 'k' is the particular term number. In this case, the middle term is T⁵₍₁ = ¹⁰C⁴ (3x)¹⁰−⁴ (∗2y)⁴
To simplify, apply the Division of Exponentials and Cubing of Exponentials principles where needed, as well as the combination formula for the coefficients. The calculation of the actual numerical coefficient is more complex and involves the combination of the coefficients (10 choose 4), the powers of the terms, and the combination of their signs. The coefficient of the middle term would ultimately be a combination of these factors.