Question

Simplify `[(6(3^(n+1)))/ (3^(n(n-1)))] /[(2(9)^(n+1)) / ((2^(n-1))^(n+1))]` ?

Answer 1

See below.

Explanation:

Assuming that the question reads

`[(6(3^(n+1)))/ (3^(n(n-1)))] /[(2(9)^(n+1)) / ((2^(n-1))^(n+1))]`

and knowing that `6=2 xx 3` and `9 = 3^2` we have

`(2 xx 3 xx 3^(n+1))/(3^(n(n-1))) = 2 xx 3^(n+1-n(n-1)+1)`

and

`(2(9)^(n+1)) / ((2^(n-1))^(n+1)) = (2 xx 3^(2(n+1)))/(2^(n^2-1)) = 3^(2n+2)/2^(n^2-2)` and finally

`2 xx 3^(n+1-n(n-1)+1) xx 2^(n^2-2) xx 3^(-2n-2) =2^(n^2-1) / 3^(n(n+2)) = (2/3)^(n^2)(1/(2 xx 3^(2n)))`