• # question_answer 1)                 Evaluate:                 (i) ${{\left\{ {{\left( \frac{1}{3} \right)}^{-1}}-{{\left( \frac{1}{4} \right)}^{-1}} \right\}}^{-1}}$                 (ii) ${{\left( \frac{5}{8} \right)}^{-7}}\,\times {{\left( \frac{8}{5} \right)}^{-4}}$

(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}}}$ $=\frac{{{5}^{-7}}}{{{5}^{-4}}}\times \frac{{{8}^{-4}}}{{{8}^{-7}}}$ $={{5}^{(-7)-(-4)}}\times \,{{8}^{(-4)-(-7)}}$ $={{5}^{-7+4}}\times {{8}^{-4+7}}$ $={{5}^{-3}}\times {{8}^{3}}=\frac{1}{{{5}^{3}}}\,\times {{8}^{3}}$ $=\frac{{{8}^{3}}}{{{5}^{3}}}\,=\frac{512}{125}$