question_answer 6)

                Evaluate:                 (i) \[{{\left\{ {{\left( \frac{1}{3} \right)}^{-1}}-{{\left( \frac{1}{4} \right)}^{-1}} \right\}}^{-1}}\]                 (ii) \[{{\left( \frac{5}{8} \right)}^{-7}}\,\times {{\left( \frac{8}{5} \right)}^{-4}}\]

Answer:

                (i) \[{{\left\{ {{\left( \frac{1}{3} \right)}^{-1}}-{{\left( \frac{1}{4} \right)}^{-1}} \right\}}^{-1}}\] \[{{\left\{ {{\left( \frac{1}{3} \right)}^{-1}}-{{\left( \frac{1}{4} \right)}^{-1}} \right\}}^{-1}}\,={{\left( \frac{{{1}^{-1}}}{{{3}^{-1}}}-\frac{{{1}^{-1}}}{{{4}^{-1}}} \right)}^{-1}}\] \[={{\left( \frac{{{3}^{1}}}{{{1}^{1}}}-\frac{{{4}^{1}}}{{{1}^{1}}} \right)}^{-1}}\] \[={{\left( \frac{3}{1}-\frac{4}{1} \right)}^{-1}}\,={{(3-4)}^{-1}}\] \[={{(-1)}^{-1}}\,=\frac{1}{{{(-1)}^{1}}}\] \[=\frac{1}{(-1)}=-1\] (ii) \[{{\left( \frac{5}{8} \right)}^{-7}}\,\times {{\left( \frac{8}{5} \right)}^{-4}}\] \[{{\left( \frac{5}{8} \right)}^{-7}}\,\times {{\left( \frac{8}{5} \right)}^{-4}}\]\[=\frac{{{5}^{-7}}}{{{8}^{-7}}}\,\times \frac{{{8}^{-4}}}{{{5}^{-4}}}\] \[=\frac{{{5}^{-7}}}{{{5}^{-4}}}\times \frac{{{8}^{-4}}}{{{8}^{-7}}}\] \[={{5}^{(-7)-(-4)}}\times \,{{8}^{(-4)-(-7)}}\] \[={{5}^{-7+4}}\times {{8}^{-4+7}}\] \[={{5}^{-3}}\times {{8}^{3}}=\frac{1}{{{5}^{3}}}\,\times {{8}^{3}}\] \[=\frac{{{8}^{3}}}{{{5}^{3}}}\,=\frac{512}{125}\]