Question

Simplify the following:

`(6^(2n + 6) - 6^3 * 36^(n + 1))/(6^(n + 2))^2`

Answer 1

Given: `(6^(2n + 6) - 6^3 * 36^(n + 1))/(6^(n + 2))^2`

Step-wise calculation:

1. Rewrite 36^{n + 1} as

`(6^2)^(n + 1) = 6^(2(n + 1))` 

`(6^2)^(n + 1) = 6^(2n + 2)`

2. Substitute into the numerator:

`6^(2n + 6) - 6^3 xx 6^(2n + 2) = 6^(2n + 6) - 6^(3 + 2n + 2)`

`6^(2n + 6) - 6^3 xx 6^(2n + 2) = 6^(2n + 6) - 6^(2n + 5)`

3. Factor out `6^(2n + 5)` from numerator:

`6^(2n + 5)(6^1 - 1) = 6^(2n + 5)(6 - 1)`

`6^(2n + 5)(6^1 - 1) = 6^(2n + 5) xx 5`

4. Denominator:

`(6^(n + 2))^2 = 6^(2(n + 2))`

`(6^(n + 2))^2 = 6^(2n + 4)`

5. Divide numerator by denominator:

`(6^(2n + 5) xx 5)/(6^(2n + 4)) = 5 xx 6^((2n + 5) - (2n + 4))`

`(6^(2n + 5) xx 5)/(6^(2n + 4)) = 5 xx 6^1`

`(6^(2n + 5) xx 5)/(6^(2n + 4)) = 5 xx 6`

`(6^(2n + 5) xx 5)/(6^(2n + 4)) = 30`