(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) \[\frac{{{3}^{-5}}\times {{10}^{-3}}\times 125}{{{5}^{-7}}\times {{6}^{-5}}}=\frac{{{5}^{7}}\times {{6}^{5}}\times (5\times 5\times 5)}{{{3}^{5}}\times {{10}^{5}}}\] \[=\frac{{{5}^{7}}\times {{(2\times 3)}^{5}}\times {{5}^{3}}}{{{3}^{5}}\times {{(2\times 5)}^{5}}}\] \[=\frac{{{5}^{7}}\times {{2}^{5}}\times {{3}^{5}}\times {{5}^{3}}}{{{3}^{5}}\times {{2}^{5}}\times {{5}^{5}}}\] \[=({{5}^{7}}\times {{5}^{3}}\times {{5}^{-5}})\times ({{2}^{5}}\times {{2}^{-5}})\times ({{3}^{5}}\times {{3}^{-5}})\] \[={{5}^{7+3-5}}\times {{2}^{5-5}}\times {{3}^{5-5}}\] \[={{5}^{10-5}}\times 2{}^\circ \times 3{}^\circ\] \[={{5}^{5}}\].
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