question_answer

Simplify and express each of the following as exponential form:

(a) \[\frac{{{2}^{3}}\times {{3}^{4}}\times 4}{3\times 32}\]

(b)\[[{{({{5}^{2}})}^{3}}\times {{5}^{4}}]\div {{5}^{7}}\]

(c) \[{{25}^{4}}\div {{5}^{3}}\]

 

Answer:

(a) \[\frac{{{2}^{3}}\times {{3}^{4}}\times 4}{3\times 32}=\frac{{{2}^{3}}\times {{3}^{4}}\times {{2}^{2}}}{3\times {{2}^{5}}}\]

\[\left[ \therefore 4=2\times 2={{2}^{2}},32=2\times 2\times 2\times 2\times 2={{2}^{5}} \right]\]

=\[\frac{{{2}^{3+2}}\times {{3}^{4}}}{{{3}^{1}}\times {{2}^{5}}}=\frac{{{2}^{5}}\times {{3}^{4}}}{{{3}^{1}}\times {{2}^{5}}}\]

\[={{2}^{55}}\times {{3}^{41}}\]

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

\[=1\times {{3}^{3}}={{3}^{3}}\]

(b) \[\left[ {{\left( {{5}^{2}} \right)}^{3}}\times {{5}^{4}} \right)]\div {{5}^{7}}=\left( {{5}^{2\times 3}}\times {{5}^{4}} \right)\div {{5}^{7}}\]

\[=\left( {{5}^{6}}\times {{5}^{4}} \right)\div {{5}^{7}}\]

\[={{5}^{6+4}}\div {{5}^{7}}={{5}^{10}}\div {{5}^{7}}\]

\[=\text{ }{{5}^{107}}={{5}^{3}}\]

(c) \[{{25}^{4}}\div {{5}^{3}}={{\left( {{5}^{2}} \right)}^{4}}\div {{5}^{3}}={{5}^{2\times 4}}\div {{5}^{3}}\]

\[={{5}^{8}}\div {{5}^{3}}={{5}^{83}}={{5}^{5}}\]