Question

Simplify:

`(9^(3n) xx 3^(n + 1))/(27^(n + 1) xx 81^n)`

Answer 1

Given,

`(9^(3n) xx 3^(n + 1))/(27^(n + 1) xx 81^n)`

We need to simplify the given terms.

Thus, `(9^(3n) xx 3^(n + 1))/(27^(n + 1) xx 81^n)`

= `((3^2)^(3n) xx 3^(n + 1))/((3^3)^(n + 1) xx (3^4)^n)`  

= `((3)^(6n) xx 3^(n + 1))/((3)^(3n + 3) xx (3)^(4n)`  ...[∴ (aⁿ)^m = a^nm]

= `((3)^(6n + n + 1))/((3)^(3n + 3 + 4n))`  ...[∴ aⁿ × a^m = a^{n + m}]

= `(3)^(6n + n + 1 - 3n - 3 - 4n)`

= `(3)^(7n + 1 - 7n - 3)`

= `(3)^-2`

= `1/3^2`

= 1/9`

Hence, the required is `1/9`.