Question

Evaluate:

`(25^(n + 1) * 5^n - 125^n)/(5^(3n) * 2^3)`

Answer 1

Given,

`(25^(n + 1) * 5^n - 125^n)/(5^(3n) * 2^3)`

We need to simplify the given expression.

Thus, `(25^(n + 1) xx 5^n - 125^n)/(5^(3n) xx 2^3)`

⇒ `((5^2)^(n + 1) xx 5^n - (5^3)^n)/(5^(3n) xx 2^3)`

⇒ `((5^2)^(2n + 2) xx 5^n - (5)^(3n))/(5^(3n) xx 2^3)`  ...[∴ (aⁿ)^m = a^nm]

⇒ `((5)^(2n + 2 + n) - (5)^(3n))/(5^(3n) xx 2^3)`  ...[∴ aⁿ × a^m = a^{n + m}]

⇒ `((5)^(3n + 2) - (5)^(3n))/(5^(3n) xx 8)`

⇒ `((5)^(3n) xx (5)^2 - (5)^(3n))/(5^(3n) xx 8)`  ...[∴ aⁿ × a^m = a^{n + m}]

Now, taking out the common term and simplifying the expression by cancelling out the same term we get,

⇒ `((5)^(3n) (5)^2 - 1)/(5^(3n) xx 8)`

⇒ `(25 - 1)/8`

= `24/8`

= 3

Hence, the required is 3.