Question

Evaluate:

`(x^(5+n)(x^2)^(3n+1))/x^(7n-2)`

Answer 1

`(x^(5+n)(x^2)^(3n+1))/x^(7n-2)`

Step 1: Simplify `(x^2)^(3n+1)`

Using the rule (a^m)ⁿ = a^m⋅n:

(x²)³ⁿ⁺¹ = x^2(3n+1) = x⁶ⁿ⁺².

Step 2: Multiply x⁵⁺ⁿ and x⁶ⁿ⁺²

Using the product rule (a^m⋅aⁿ = a^m+n):

x⁵⁺ⁿ⋅x⁶ⁿ⁺² = x^(5+n)+(6n+2) = x^5+n+6n+2 = x⁷⁺⁷ⁿ.

Step 3: Divide by x⁷ⁿ⁻²

Using the quotient rule `a^m/a^n = a^(m-n)`:

`x^(7+7n)/x^(7n-2)`

`= x^((7+7n)−(7n−2))`

`= x^(7+7n−7n+2)`

`= x^(7+2)`

`= x^9`