Simplify and express each of the following as a rational number :

(i) $\dfrac{10^2 \times 15^3}{2^2 \times 3 \times 5^5 \times 6^4}$

(ii) $\dfrac{3^5 \times 25 \times 10^5}{5^7 \times 6^5}$

(i) We have:

$\phantom{=} \dfrac{10^2 \times 15^3}{2^2 \times 3 \times 5^5 \times 6^4} \\[1em] = \dfrac{(2 \times 5)^2 \times (3 \times 5)^3}{2^2 \times 3^1 \times 5^5 \times (2 \times 3)^4} \quad \text{[Prime factorizing bases]} \\[1em] = \dfrac{2^2 \times 5^2 \times 3^3 \times 5^3}{2^2 \times 3^1 \times 5^5 \times 2^4 \times 3^4} \\[1em] = \dfrac{2^2 \times 3^3 \times 5^{2+3}}{2^{2+4} \times 3^{1+4} \times 5^5} \\[1em] = \dfrac{2^2 \times 3^3 \times 5^5}{2^6 \times 3^5 \times 5^5} \\[1em] = \dfrac{2^{2-6}}{3^{5-3} \times 5^{5-5}} \\[1em] = \dfrac{2^{-4}}{3^2 \times 5^0} \quad \text{[Using law of exponents]} \\[1em] = \dfrac{1}{2^4 \times 3^2 \times 1} \quad [2^{-4} = \dfrac{1}{2^4}] \\[1em] = \dfrac{1}{16 \times 9} \\[1em] = \dfrac{1}{144}$

Hence, the answer is $\dfrac{1}{144}$

(ii) We have:

$\phantom{=} \dfrac{3^5 \times 25 \times 10^5}{5^7 \times 6^5} \\[1em] = \dfrac{3^5 \times 5^2 \times (2 \times 5)^5}{5^7 \times (2 \times 3)^5} \quad \text{[Prime factorizing bases]} \\[1em] = \dfrac{3^5 \times 5^2 \times 2^5 \times 5^5}{5^7 \times 2^5 \times 3^5} \\[1em] = \dfrac{2^5 \times 3^5 \times 5^{2+5}}{2^5 \times 3^5 \times 5^7} \\[1em] = \dfrac{2^5 \times 3^5 \times 5^7}{2^5 \times 3^5 \times 5^7} \\[1em] = 2^{5-5} \times 3^{5-5} \times 5^{7-7} \quad \text{[Using law of exponents]} \\[1em] = 2^0 \times 3^0 \times 5^0 \\[1em] = 1 \times 1 \times 1 \\[1em] = 1$

Hence, the answer is 1