Simplify the following:
5n+3−6×5n+19×5n−22×5n\dfrac{5^{n + 3} - 6 \times 5^{n + 1}}{9 \times 5^n - 2^2 \times 5^n}9×5n−22×5n5n+3−6×5n+1
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Given,
⇒5n+3−6×5n+19×5n−22×5n=5n.53−6.5n.515n(9−22)=5n.(53−30)5n(9−4)=125−305=955=19.\Rightarrow \dfrac{5^{n + 3} - 6 \times 5^{n + 1}}{9 \times 5^n - 2^2 \times 5^n} = \dfrac{5^n.5^3 - 6.5^n.5^1}{5^n(9 - 2^2)} \\[1em] = \dfrac{5^n.(5^3 - 30)}{5^n(9 - 4)} \\[1em] = \dfrac{125 - 30}{5} \\[1em] = \dfrac{95}{5} = 19.⇒9×5n−22×5n5n+3−6×5n+1=5n(9−22)5n.53−6.5n.51=5n(9−4)5n.(53−30)=5125−30=595=19.
Hence, 5n+3−6×5n+19×5n−22×5n\dfrac{5^{n + 3} - 6 \times 5^{n + 1}}{9 \times 5^n - 2^2 \times 5^n}9×5n−22×5n5n+3−6×5n+1 = 19.
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