Question

Evaluate:

`(343^n + 49^n * 7^(n + 2))/((5 * 7^n)^3 - 25 xx 7^(3n))`

Answer 1

Given,

`(343^n + 49^n * 7^(n + 2))/((5 * 7^n)^3 - 25 xx 7^(3n))`

We need to simplify the given expression.

Thus, `(343^n + 49^n xx 7^(n + 2))/((5 xx 7^n)^3 - 25 xx 7^(3n))`

⇒ `((7^3)^n + (7^2)^n xx 7^(n + 2))/((5 xx 7^n)^3 - 5^2 xx 7^(3n))`

⇒ `((7)^(3n) + (7)^(2n) xx 7^(n + 2))/(5^3 xx 7^(3n) - 5^2 xx 7^(3n))`  ...[∴ (aⁿ)^m = a^nm]

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

⇒ `((7)^(3n) + (7)^(2n + n + 2))/(5^2 xx 7^(3n) (5 - 1))`  ...[∴ aⁿ × a^m = a^{n + m}]

⇒ `((7)^(3n) + (7)^(3n + 2))/(5^2 xx 7^(3n) xx 4)`

⇒ `((7)^(3n) + (7)^(3n) xx 7^2)/(5^2 xx 7^(3n) xx 4)`   ...[∴ aⁿ × a^m = a^{n + m}]

⇒ `((7)^(3n) + (7)^(3n) xx 49)/(25 xx 7^(3n) xx 4)`

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

= `((7)^(3n)(1 + 49))/(25 xx 7^(3n) xx 4)`

= `50/(25 xx 4)`

= `2/4`

= `1/2`

Hence, the required is `1/2`.