question_answer

(a)                                                         \[{{(-9)}^{3}}\div {{(-9)}^{8}}\]

(b)                    \[\frac{{{3}^{-5}}\times {{10}^{-5}}\times 125}{{{5}^{-7}}\times {{6}^{-5}}}\]

Answer:

(a)          \[{{\left( 9 \right)}^{3}}{{\left( \text{ }9 \right)}^{8}}=\text{ }{{\left( \text{ }9 \right)}^{8}}\]

\[=\frac{{{(-9)}^{3}}}{{{(-9)}^{8}}}\]

\[={{(-9)}^{3-8}}\]

\[={{(-9)}^{-5}}\]

\[=\frac{1}{{{(-9)}^{5}}}\]

(b) \[\frac{{{3}^{-5}}\times {{10}^{-3}}\times 125}{{{5}^{-7}}\times {{6}^{-5}}}=\frac{{{5}^{7}}\times {{6}^{5}}\times (5\times 5\times 5)}{{{3}^{5}}\times {{10}^{5}}}\]

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

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

\[=({{5}^{7}}\times {{5}^{3}}\times {{5}^{-5}})\times ({{2}^{5}}\times {{2}^{-5}})\times ({{3}^{5}}\times {{3}^{-5}})\]

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

\[={{5}^{10-5}}\times 2{}^\circ \times 3{}^\circ\]

\[={{5}^{5}}\].