Simplify:

Solution:

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

$=\frac{3^{\mathrm{n}} \times 9^{\mathrm{n}} \times 9}{\frac{3^{\mathrm{n}}}{3} \times \frac{9^{\mathrm{n}}}{9}}=9 \times 3 \times 9=243$

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

$=\frac{\left(5 \times 25^{\mathrm{n}} \times 25\right)-\left(25 \times 25^{\mathrm{n}}\right)}{\left(5 \times 25^{\mathrm{n}} \times 125\right)\left(25^{\mathrm{n}} \times 25\right)}$

$=\frac{25^{\mathrm{n}} \times 25(5-1)}{25^{\mathrm{n}} \times 25(25-1)}$

$=\frac{4}{24}=\frac{1}{6}$

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

$=\frac{\left(5^{\mathrm{n}+3}\right)-\left(6 \times 5^{\mathrm{n}+1}\right)}{\left(9 \times 5^{\mathrm{n}}\right)-\left(2^{2} \times 5^{\mathrm{n}}\right)}$

$=\frac{\left(5^{\mathrm{n}} \times 5^{3}\right)-\left(6 \times 5^{\mathrm{n}} \times 5\right)}{\left(9 \times 5^{\mathrm{n}}\right)-\left(2^{2} \times 5^{\mathrm{n}}\right)}$

$=\frac{5^{n}(125-30)}{5^{n}(9-4)}$

$=\frac{95}{5}=19$

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

$=\frac{\left(6 \times 8^{\mathrm{n}} \times 8\right)+\left(16 \times 8^{\mathrm{n}} \times \frac{1}{4}\right)}{\left(10 \times 8^{\mathrm{n}} \times 2\right)-7 \times(8)^{\mathrm{n}}}$

$=\frac{8^{n}(48+4)}{8^{n}(20-7)}$

$=\frac{52}{13}=4$