question_answer 7)

                Simplify:                 (i) \[\frac{25\times {{t}^{-4}}}{{{5}^{-3}}\times 10\times {{t}^{-8}}}\,(t\ne \,0)\]                 (ii) \[\frac{{{3}^{-5}}\times {{10}^{-5}}\times 125}{{{5}^{-7}}\,\times {{6}^{-5}}}\]

Answer:

                (i) \[\frac{25\times {{t}^{-4}}}{{{5}^{-3}}\times 10\times {{t}^{-8}}}\,(t\ne \,0)\] \[\frac{25\times {{t}^{-4}}}{{{5}^{-3}}\times 10\times {{t}^{-8}}}\,=\frac{25\times \frac{1}{{{t}^{4}}}}{\frac{1}{{{5}^{3}}}\times 10\times \frac{1}{{{t}^{8}}}}\] \[=\frac{\frac{25}{{{t}^{4}}}}{\frac{1}{125}\times 10\times \frac{1}{{{t}^{8}}}}\] \[=\frac{\frac{25}{{{t}^{4}}}}{\frac{2}{25{{t}^{8}}}}\] \[=\frac{25}{{{t}^{4}}}\times \frac{25\,{{t}^{8}}}{2}\] \[=\,\frac{625\,\,{{t}^{8-4}}}{2}\] \[=\frac{625}{2}{{t}^{4}}\]                 (ii) \[\frac{{{3}^{-5}}\times {{10}^{-5}}\times 125}{{{5}^{-7}}\,\times {{6}^{-5}}}\]                 \[\frac{{{3}^{-5}}\times {{10}^{-5}}\times 125}{{{5}^{-7}}\,\times {{6}^{-5}}}\] \[=\frac{{{3}^{-5}}\times {{(2\times 5)}^{-5}}\times (5\times 5\times 5)}{{{5}^{-7}}\times {{(2\times 3)}^{-5}}}\] \[=\frac{{{3}^{-5}}\times {{2}^{-5}}\,\times {{5}^{-5}}\,\times {{5}^{3}}}{{{5}^{-7}}\,\times {{2}^{-5}}\times {{3}^{-5}}}\] \[=\frac{{{5}^{-5}}\,\times {{5}^{3}}}{{{5}^{-7}}}\] \[=\frac{{{5}^{(-5)+3}}}{{{5}^{-7}}}\] \[=\frac{{{5}^{-2}}}{{{5}^{-7}}}\] \[={{5}^{(-2)-(-7)}}\] \[=5{{\,}^{-2+7}}={{5}^{5}}\]