Simplify :

$\dfrac{5^{2(n + 6)} \times (25)^{-7 + 2n}}{(125)^{2n}}$

Given,

$\dfrac{5^{2(n + 6)} \times (25)^{-7 + 2n}}{(125)^{2n}}$

Simplifying the expression :

$\Rightarrow \dfrac{5^{2(n + 6)} \times (25)^{-7 + 2n}}{(125)^{2n}} \\[1em] \Rightarrow \dfrac{5^{2n + 12} \times [(5)^2]^{-7 + 2n}}{[(5)^3]^{2n}} \\[1em] \Rightarrow \dfrac{5^{2n + 12} \times 5^{2(-7 + 2n)}}{(5)^{6n}} \\[1em] \Rightarrow \dfrac{5^{2n + 12} \times 5^{-14 + 4n}}{(5)^{6n}} \\[1em] \Rightarrow \dfrac{5^{2n + 12 + (-14 + 4n)}}{(5)^{6n}} \\[1em] \Rightarrow \dfrac{5^{6n - 2}}{(5)^{6n}} \\[1em] \Rightarrow 5^{6n - 2 - 6n} \\[1em] \Rightarrow 5^{-2} \\[1em] \Rightarrow \Big(\dfrac{1}{5}\Big)^{2} \\[1em] \Rightarrow \dfrac{1}{25}.$

Hence, $\dfrac{5^{2(n + 6)} \times (25)^{-7 + 2n}}{(125)^{2n}} = \dfrac{1}{25}$.