Simplify :

$\dfrac{7^{(2n + 3)} - (49)^{n + 2}}{\Big[(343)^{n + 1}\Big]^{\dfrac{2}{3}}}$

Given,

$\dfrac{7^{(2n + 3)} - (49)^{n + 2}}{\Big[(343)^{n + 1}\Big]^{\dfrac{2}{3}}}$

Simplifying the expression :

$\Rightarrow \dfrac{7^{(2n + 3)} - (49)^{n + 2}}{\Big[(343)^{n + 1}\Big]^{\dfrac{2}{3}}} \\[1em] \Rightarrow \dfrac{7^{(2n + 3)} - [(7)^2]^{n + 2}}{\Big[[(7)^3]^{n + 1}\Big]^{\dfrac{2}{3}}} \\[1em] \Rightarrow \dfrac{7^{(2n + 3)} - (7)^{2n + 4}}{7^{3 \times (n + 1) \times \dfrac{2}{3}}} \\[1em] \Rightarrow \dfrac{7^{(2n + 3)} - 7^{2n + 3 + 1}}{7^{{(n + 1)} \times 2}} \\[1em] \Rightarrow \dfrac{7^{(2n + 3)}(1 - 7)}{(7)^{2n + 2}} \\[1em] \Rightarrow 7^{(2n + 3) - (2n + 2)} \times -6 \\[1em] \Rightarrow 7^{2n + 3 - 2n - 2} \times -6 \\[1em] \Rightarrow 7^1 \times -6 \\[1em] \Rightarrow -42.$

Hence, $\dfrac{7^{(2n + 3)} - (49)^{n + 2}}{\Big[(343)^{n + 1}\Big]^{\dfrac{2}{3}}} = - 42$.