Solutions :
1. [tex]\bf \dfrac{18m^2u}{16n^3v^2} \div \frac{24m}{15nu^3} \times\frac{8n^2v^3}{30m^3u} \\[/tex]
[tex]\tt : \implies \dfrac{18m^2u}{16n^3v^2} \times \dfrac{15nu^3}{24m} \times\dfrac{8n^2v^3}{30m^3u}[/tex]
[tex]\tt : \implies \dfrac{18\times m\times m \times u}{16\times n\times n \times n \times v\times v} \times \dfrac{15\times n\times u\times u \times u}{24\times m} \times\dfrac{8\times n\times n \times v\times v \times v}{30\times m\times m \times m \times u}[/tex]
[tex]\tt : \implies \dfrac{18\times \cancel m\times \cancel m \times \cancel u}{16\times \cancel n\times \cancel n \times \cancel n \times \cancel v\times \cancel v} \times \dfrac{15\times \cancel n\times u\times u \times u}{24\times \cancel m} \times\dfrac{8\times \cancel n\times \cancel n \times \cancel v\times \cancel v \times v}{30\times \cancel m\times m \times m \times \cancel u}[/tex]
[tex]\tt : \implies \dfrac{18}{16} \times \dfrac{15\times u\times u \times u}{24} \times\dfrac{8\times v}{30\times m \times m}[/tex]
[tex]\tt : \implies \cancel{\dfrac{18}{16}} \times \dfrac{\cancel{15}\times u\times u \times u}{\cancel{24}} \times\dfrac{\cancel{8}\times v}{\cancel{30}\times m \times m}[/tex]
[tex]\tt : \implies \dfrac{9}{8} \times \dfrac{1\times u\times u \times u}{3} \times\dfrac{1\times v}{2\times m \times m}[/tex]
[tex]\tt : \implies \dfrac{\cancel{9}}{8} \times \dfrac{u^3}{\cancel{3}} \times\dfrac{v}{2m^2}[/tex]
[tex]\tt : \implies \dfrac{3}{8} \times u^3 \times\dfrac{v}{2m^2}[/tex]
[tex]\tt : \implies \dfrac{3\times u^3\times v}{8\times 2m^2}[/tex]
[tex]\tt : \implies \dfrac{3u^3v}{16m^2}\\[/tex]
[tex]\boxed{\bf Hence, \: answer \: is \dfrac{3u^3v}{16m^2}}[/tex]
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2. [tex]\bf \dfrac{24a^2b^3c}{9bc^3} \div \dfrac{4a^5bc^3}{27a^3b^2c}\\[/tex]
[tex]\tt : \implies \dfrac{24a^2b^3c}{9bc^3} \times \dfrac{27a^3b^2c}{4a^5bc^3}[/tex]
[tex]\tt : \implies \dfrac{24\times a\times a \times b\times b\times b\times c}{9\times b\times c\times c\times c} \times \dfrac{27\times a\times a\times a\times b\times b\times c}{4\times a\times a\times a\times a\times a\times b\times c\times c\times c}[/tex]
[tex]\tt : \implies \dfrac{24\times \cancel a\times \cancel a \times \cancel b\times \cancel b\times b\times \cancel c}{9\times \cancel b\times \cancel c\times c\times c} \times \dfrac{27\times \cancel a\times \cancel a\times \cancel a\times b\times b\times \cancel c}{4\times \cancel a\times \cancel a\times \cancel a\times \cancel a\times \cancel a\times \cancel b\times \cancel c\times c\times c}[/tex]
[tex]\tt : \implies \dfrac{\cancel{24}\times b}{\cancel{9}\times c\times c} \times \dfrac{\cancel{27}\times b\times b}{\cancel{4}\times c\times c}[/tex]
[tex]\tt : \implies \dfrac{6\times b}{1\times c\times c} \times \dfrac{3\times b\times b}{1\times c\times c}[/tex]
[tex]\tt : \implies \dfrac{6b}{c^2} \times \dfrac{3b^2}{c^2}[/tex]
[tex]\tt : \implies \dfrac{6b\times 3b^2}{c^2\times c^2}[/tex]
[tex]\tt : \implies \dfrac{18b^3}{c^4}\\[/tex]
[tex]\boxed{\bf Hence, \: answer \: is \dfrac{18b^3}{c^4}}[/tex]
