question_answer

If \[\frac{5m\times {{5}^{3}}\times {{5}^{-2}}}{{{5}^{-5}}}={{5}^{12}},\] find m.

Answer:

\[\frac{{{5}^{m}}\times {{5}^{3}}\times {{5}^{-2}}}{{{5}^{-5}}}={{5}^{12}}\]

\[{{5}^{m}}\times {{5}^{3}}-{{5}^{-2}}\times {{5}^{5}}={{5}^{12}}\]

\[{{5}^{m}}\times {{5}^{3}}^{-2+5}={{5}^{12}}\]

\[{{5}^{m}}\times {{5}^{6}}={{5}^{12}}\]

\[{{5}^{m+6}}={{5}^{12}}\]

On comparing powers m + 6 = 12

\[m=126\]

m = 6