Simplify the following:
Question:

Simplify the following:

(i) $\frac{3^{n} \times 9^{n+1}}{3^{n-1} \times 9^{n-1}}$

(ii) $\frac{5 \times 25^{n+1}-25 \times 5^{2 n}}{5 \times 5^{2 n+3}-25^{n+1}}$

(iii) $\frac{5^{n+3}-6 \times 5^{n+1}}{9 \times 5^{x}-2^{2} \times 5^{n}}$

(iv) $\frac{6(8)^{n+1}+16(2)^{3 n-2}}{10(2)^{3 n+1}-7(8)^{n}}$

Solution:

(i)

$\frac{3^{n} \times 9^{n+1}}{3^{n-1} \times 9^{n-1}}$

$=\frac{3^{n} \times\left(3^{2}\right)^{(n+1)}}{3^{n-1} \times\left(3^{2}\right)^{n-1}}$

$=\frac{3^{n} \times 3^{2 n+2}}{3^{n-1} \times 3^{2 n-2}}$

$=\frac{3^{n+2 n+2}}{3^{n-1+2 n-2}}$

$=\frac{3^{3 n+2}}{3^{3 n-3}}$

$=3^{3 n+2-3 n+3}$

$=3^{5}$

$=243$

(ii)

$\frac{5 \times 25^{n+1}-25 \times 5^{2 n}}{5 \times 5^{2 n+3}-25^{n+1}}$

$=\frac{5 \times\left(5^{2}\right)^{n+1}-\left(5^{2}\right) \times 5^{2 n}}{5 \times 5^{2 n+3}-\left(5^{2}\right)^{n+1}}$

$=\frac{5 \times\left(5^{2 n+2}\right)-\left(5^{2}\right) \times 5^{2 n}}{5 \times 5^{2 n+3}-\left(5^{2 n+2}\right)}$

$=\frac{5^{1+2 n+2}-5^{2+2 n}}{5^{1+2 n+3}-5^{2 n+2}}$

$=\frac{5^{2 n+3}-5^{2+2 n}}{5^{2 n+4}-5^{2 n+2}}$

$=\frac{5^{2+2 n}(5-1)}{5^{2 n+2}\left(5^{2}-1\right)}$

$=\frac{4}{24}$

$=\frac{1}{6}$

(iii)

$\frac{5^{n+3}-6 \times 5^{n+1}}{9 \times 5^{n}-2^{2} \times 5^{n}}$

$=\frac{5^{n+1}\left(5^{2}-6\right)}{5^{n}\left(9-2^{2}\right)}$

$=\frac{5^{n} \times 5 \times(25-6)}{5^{n}(9-4)}$

$=\frac{5 \times 19}{5}$

$=19$

(iv)

$\frac{6(8)^{n+1}+16(2)^{3 n-2}}{10(2)^{3 n+1}-7(8)^{n}}$

$=\frac{6\left(2^{3}\right)^{n+1}+16(2)^{3 n-2}}{10(2)^{3 n+1}-7\left(2^{3}\right)^{n}}$

$=\frac{6\left(2^{3 n+3}\right)+16(2)^{3 n-2}}{10(2)^{3 n+1}-7\left(2^{3 n}\right)}$

$=\frac{6 \times 2^{3 n}\left(2^{3}\right)+16(2)^{3 n} 2^{-2}}{10(2)^{3 n}\left(2^{1}\right)-7\left(2^{3 n}\right)}$

$=\frac{2^{3 n}(48+4)}{2^{3 n}(20-7)}$

$=\frac{52}{13}$

$=4$

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