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

(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) $\frac{{{3}^{-5}}\times {{10}^{-5}}\times 125}{{{5}^{-7}}\,\times {{6}^{-5}}}$                 $\frac{{{3}^{-5}}\times {{10}^{-5}}\times 125}{{{5}^{-7}}\,\times {{6}^{-5}}}$ $=\frac{{{3}^{-5}}\times {{(2\times 5)}^{-5}}\times (5\times 5\times 5)}{{{5}^{-7}}\times {{(2\times 3)}^{-5}}}$ $=\frac{{{3}^{-5}}\times {{2}^{-5}}\,\times {{5}^{-5}}\,\times {{5}^{3}}}{{{5}^{-7}}\,\times {{2}^{-5}}\times {{3}^{-5}}}$ $=\frac{{{5}^{-5}}\,\times {{5}^{3}}}{{{5}^{-7}}}$ $=\frac{{{5}^{(-5)+3}}}{{{5}^{-7}}}$ $=\frac{{{5}^{-2}}}{{{5}^{-7}}}$ $={{5}^{(-2)-(-7)}}$ $=5{{\,}^{-2+7}}={{5}^{5}}$