Question

Simplify the following:

`(5^(n + 5) - 6 xx 5^(n + 3))/(9 xx 5^(n + 1) - 20 xx 5^n)`

Answer 1

Given: `(5^(n + 5) - 6 xx 5^(n + 3))/(9 xx 5^(n + 1) - 20 xx 5^n)`

Step-wise calculation:

1. Factor out the common term in numerator:

`5^(n + 3)` is common in `5^(n + 5)` and `6 xx 5^(n + 3)` 

Rewrite the numerator as:

`5^(n + 3)(5^2 - 6) = 5^(n + 3)(25 - 6)`

`5^(n + 3)(5^2 - 6) = 5^(n + 3) xx 19`

2. Factor out the common term in denominator:

`5^n` is common in `9 xx 5^(n + 1)` and `20 xx 5^n` 

Rewrite denominator:

`5^n (9 xx 5^(1) - 20) = 5^n (45 - 20)`

`5^n (9 xx 5^(1) - 20) = 5^n xx 25`

3. Now the expression is:

`(5^(n + 3) xx 19)/(5^n xx 25)`

4. Simplify the powers of 5:

`(5^(n + 3))/5^n = 5^((n + 3) - n)`

`(5^(n + 3))/5^n = 5^3`

`(5^(n + 3))/5^n = 125`

5. Substitute back:

`(19 xx 125)/25 = 19 xx 125/25`

`(19 xx 125)/25 = 19 xx 5`

`(19 xx 125)/25 = 95`

`(5^(n + 5) - 6 xx 5^(n + 3))/(9 xx 5^(n + 1) - 20 xx 5^n) = 95`

Thus, the expression simplifies to 95.