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In the binomial expansion of ((2)/(x)+x^(log₈ x))⁶ for x>0, if the fourth term equals 20× 8⁷, then one possible value of x is:

1. 8³
2. 8²
3. 8
4. 8⁻²

Correct answer: 8²

Solution

T4 = C(6,3)*(2/x)^3*(x^(log_8 x))^3 = 160*x^(3*log_8 x - 3). Setting this equal to 20*8^7 = 20*2^21 gives x^(3*log_8 x - 3) = 2^18, whose solutions are x = 8^(-1) and x = 8^2. Among the choices, x = 8^2.

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