question_answer 1)

Simplify (i) \[\frac{{{({{2}^{5}})}^{2}}\times {{7}^{3}}}{{{8}^{3}}\times 7}\]                              (ii) \[\frac{25\times {{5}^{2}}\times {{t}^{8}}}{{{10}^{3}}\times {{t}^{4}}}\]                    (iii) \[\frac{{{3}^{5}}\times {{10}^{5}}\times 25}{{{5}^{7}}\times {{6}^{5}}}\]

Answer:

                (i) \[\frac{{{({{2}^{5}})}^{2}}\times {{7}^{3}}}{{{8}^{3}}\times 7}=\frac{{{2}^{5\times 2}}\times {{7}^{3}}}{{{({{2}^{3}})}^{3}}\times 7}=\frac{{{2}^{10}}\times {{7}^{3}}}{{{2}^{3\times 3}}\times 7}\]                 \[=\frac{{{2}^{10}}\times {{7}^{3}}}{{{2}^{9}}\times 7}={{2}^{10-9}}\times {{7}^{3-1}}={{2}^{1}}\times {{7}^{2}}\]                 \[=2\times 7\times 7=98\]                 (ii) \[\frac{25\times {{5}^{2}}\times {{t}^{8}}}{{{10}^{3}}\times {{t}^{4}}}=\frac{{{5}^{2}}\times {{5}^{2}}\times {{t}^{8}}}{{{(2\times 5)}^{3}}\times {{t}^{4}}}=\frac{{{5}^{2+2}}\times {{t}^{8}}}{{{2}^{3}}\times {{5}^{3}}\times {{t}^{4}}}\] \[=\frac{{{5}^{4}}\times {{t}^{8}}}{{{2}^{3}}\times {{5}^{3}}\times {{t}^{4}}}=\frac{{{5}^{4-3}}\times {{t}^{8-4}}}{{{2}^{3}}}=\frac{{{5}^{1}}\times {{t}^{4}}}{{{2}^{3}}}=\frac{5{{t}^{4}}}{8}\] (iii) \[\frac{{{3}^{5}}\times {{10}^{5}}\times 25}{{{5}^{7}}\times {{6}^{5}}}=\frac{{{3}^{5}}\times {{(2\times 5)}^{5}}\times {{5}^{2}}}{{{5}^{7}}\times {{(2\times 3)}^{5}}}\] \[=\frac{{{3}^{5}}\times {{2}^{5}}\times {{2}^{5}}\times {{2}^{5}}}{{{5}^{7}}\times {{2}^{5}}\times {{3}^{5}}}\] \[=\frac{{{3}^{5}}\times {{2}^{5}}\times {{5}^{5\times 2}}}{{{2}^{5}}\times {{3}^{5}}\times {{5}^{7}}}=\frac{{{2}^{5}}\times {{3}^{5}}\times {{5}^{7}}}{{{2}^{5}}\times {{3}^{5}}\times {{5}^{7}}}\] \[={{2}^{5-5}}\times {{3}^{5-5}}\times {{5}^{7-7}}={{2}^{0}}\times {{3}^{0}}\times {{5}^{0}}\] \[=1\times 1\times 1=1\]