question_answer

Simplify:

(A) \[\frac{{{\left( {{2}^{5}} \right)}^{2}}\times {{7}^{3}}}{{{8}^{3}}\times 7}\]

(B) \[\frac{25\times {{5}^{2}}\times {{t}^{8}}}{{{10}^{3}}\times {{t}^{4}}}\]

(c) \[\frac{{{3}^{5}}\times {{10}^{5}}\times 25}{{{5}^{7}}\times {{6}^{5}}}\]

Answer:

(a) \[\frac{{{\left( {{2}^{5}} \right)}^{2}}\times {{7}^{3}}}{{{8}^{3}}\times 7}=\frac{{{2}^{10}}\times {{7}^{3}}}{{{\left( {{2}^{3}} \right)}^{3}}\times 7}=\frac{{{2}^{10}}\times {{7}^{3}}}{{{2}^{9}}\times 7}\]

\[=\text{ }{{2}^{10}}^{9}\times {{7}^{31}}={{2}^{1}}\times {{7}^{2}}\]

= 2 × 49 = 98

(b) \[\frac{25\times {{5}^{2}}\times {{t}^{8}}}{{{10}^{3}}\times {{t}^{4}}}=\frac{{{5}^{2}}\times {{5}^{2}}\times {{t}^{8}}}{{{\left( 2\times 5 \right)}^{3}}\times {{t}^{4}}}=\frac{{{5}^{4}}\times {{t}^{8}}}{{{2}^{3}}\times {{5}^{3}}\times {{t}^{4}}}\]

\[=\frac{{{5}^{4-3}}\times {{t}^{8-4}}}{{{2}^{3}}}=\frac{5\times {{t}^{4}}}{{{2}^{3}}}=\frac{5{{t}^{4}}}{8}\]

(c) \[\frac{{{3}^{5}}\times {{10}^{5}}\times 25}{{{5}^{7}}\times {{6}^{5}}}=\frac{{{3}^{5}}\times {{(2\times 5)}^{5}}\times {{5}^{2}}}{{{5}^{7}}\times {{(2\times 3)}^{5}}}\]

\[=\frac{{{3}^{5}}\times {{2}^{5}}\times {{5}^{5}}\times {{5}^{5}}}{{{5}^{7}}\times {{2}^{5}}\times {{3}^{5}}}\]

\[=\text{ }{{3}^{55}}\times {{2}^{55}}\times {{5}^{5}}{{^{+2}}^{7}}\]

\[=\text{ }{{3}^{0}}\times {{2}^{0}}\times {{5}^{0}}\]

= 1 × 1 × 1 = 1