For what value of x is the ninth term in the expansion of `(3^(log_3 sqrt(25^(x-1) +7)) + 3^(-1/8 log_3 (5^(x-1) +1)))^10` is equal to `180`

Explanation

Given

`(3^(log_3 sqrt(25^(x-1) +7)) + 3^(-1/8 log_3 (5^(x-1) +1)))^10`

Let `5^(x−1)=t`

Then equation is


`[sqrt(t^2+7)
​+(t+1)^(−1/8)​]^10`

8th term of the expansion is `t_8`=`10_(C_8)​(t^2+7)^(2/2)(t+1)^(−1)=45xx (t^2+7)/(t+1)​`

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