question_answer 2)

                Simplify and express the result in power notation with positive exponent.                 (i) \[{{(-4)}^{5}}\div {{(-4)}^{8}}\]                              (ii) \[{{\left( \frac{1}{{{2}^{3}}} \right)}^{2}}\]                      (iii) \[{{(-3)}^{4}}\,\times {{\left( \frac{5}{3} \right)}^{4}}\]   (iv) \[({{3}^{-7}}\div {{3}^{-10}})\times {{3}^{-5}}\]                 (v) \[{{2}^{-3}}\times \,{{(-7)}^{-3}}\]

Answer:

                (i) \[{{(-4)}^{5}}\div {{(-4)}^{8}}\]                              \[{{(-4)}^{5}}\div {{(-4)}^{8}}=\frac{{{(-4)}^{5}}}{{{(-4)}^{8}}}\] \[=\frac{1}{{{(-4)}^{8-5}}}\] \[=\frac{1}{{{(-4)}^{3}}}\] (ii) \[{{\left( \frac{1}{{{2}^{3}}} \right)}^{2}}\] \[{{\left( \frac{1}{{{2}^{3}}} \right)}^{2}}=\frac{{{1}^{2}}}{{{({{2}^{3}})}^{2}}}\] \[=\frac{1}{{{2}^{3\times 2}}}\] \[=\frac{1}{{{2}^{6}}}\]                 (iii) \[{{(-3)}^{4}}\,\times {{\left( \frac{5}{3} \right)}^{4}}\]            \[{{(-3)}^{4}}\,\times {{\left( \frac{5}{3} \right)}^{4}}\]\[={{\{(-1)\,\times 3\}}^{4}}\times {{\left( \frac{5}{3} \right)}^{4}}\] \[={{(-1)}^{4}}\times {{3}^{4}}\,\times \frac{{{5}^{4}}}{{{3}^{4}}}\] \[={{(5)}^{4}}\] (iv) \[({{3}^{-7}}\div {{3}^{-10}})\times {{3}^{-5}}\] \[({{3}^{-7}}\div {{3}^{-10}})\times {{3}^{-5}}\]\[=\left( \frac{{{3}^{-7}}}{{{3}^{-10}}} \right)\,\times \frac{1}{{{3}^{5}}}\] \[={{3}^{(-7)-\,(-10)}}\times \frac{1}{{{3}^{5}}}\] \[={{3}^{-7\,+10}}\times \frac{1}{{{3}^{5}}}\] \[=\frac{{{3}^{3}}}{{{3}^{5}}}\] \[=\frac{1}{{{3}^{5-3}}}\] \[=\frac{1}{{{(3)}^{2}}}\]                 (v) \[{{2}^{-3}}\times \,{{(-7)}^{-3}}\]                 \[{{2}^{-3}}\times \,{{(-7)}^{-3}}\] \[=\frac{1}{{{2}^{3}}}\times \,\frac{1}{{{(-7)}^{3}}}\]                 \[=\,\frac{1}{{{[2\times (-7)]}^{3}}}\]                 \[=\frac{1}{{{(-14)}^{3}}}\]