question_answer 3)

                Find the value of:                 (i) \[({{3}^{0}}+{{4}^{-1}})\,\times {{2}^{2}}\]                                      (ii) \[({{2}^{-1}}\times {{4}^{-1}})\div {{2}^{-2}}\]                 (iii) \[{{\left( \frac{1}{2} \right)}^{-2}}\,+{{\left( \frac{1}{3} \right)}^{-2}}\,+{{\left( \frac{1}{4} \right)}^{-2}}\]                                (iv) \[{{({{3}^{-1}}+{{4}^{-1}}\,+{{5}^{-1}})}^{0}}\]                 (v) \[{{\left\{ {{\left( \frac{-2}{3} \right)}^{-2}} \right\}}^{2}}\]

Answer:

                (i) \[({{3}^{0}}+{{4}^{-1}})\,\times {{2}^{2}}\]                 \[({{3}^{0}}+{{4}^{-1}})\,\times {{2}^{2}}\]\[=\left( 1+\frac{1}{4} \right)\times 4\]                 \[=\frac{5}{4}\times 4\]                 = 5                 (ii) \[({{2}^{-1}}\times {{4}^{-1}})\div {{2}^{-2}}\]                 \[({{2}^{-1}}\times {{4}^{-1}})\div {{2}^{-2}}\]\[=\{{{2}^{-1}}\,\times {{({{2}^{2}})}^{-1}}\}\div {{2}^{-2}}\] \[=\{{{2}^{-1}}\times {{2}^{2\,\times \,(-1)}}\}\,\div \,{{2}^{-2}}\] \[=({{2}^{-1}}\times {{2}^{-2}})\,\div {{2}^{-2}}\] \[={{2}^{(-1)\,+(-2)}}\div \,{{2}^{-2}}\] \[={{2}^{-3}}\div \,{{2}^{-2}}\] \[=\frac{{{2}^{-3}}}{{{2}^{-2}}}\] \[=\frac{1}{{{2}^{(-2)\,-(-3)}}}\] \[=\frac{1}{{{2}^{-2+3}}}\] \[=\frac{1}{{{2}^{1}}}\] \[=\frac{1}{2}\]                 (iii) \[{{\left( \frac{1}{2} \right)}^{-2}}\,+{{\left( \frac{1}{3} \right)}^{-2}}\,+{{\left( \frac{1}{4} \right)}^{-2}}\]                 \[{{\left( \frac{1}{2} \right)}^{-2}}\,+{{\left( \frac{1}{3} \right)}^{-2}}\,+{{\left( \frac{1}{4} \right)}^{-2}}\] \[=\frac{{{1}^{-2}}}{{{2}^{-2}}}\,+\frac{{{1}^{-2}}}{{{3}^{-2}}}\,+\frac{{{1}^{-2}}}{{{4}^{-2}}}\] \[=\frac{{{2}^{2}}}{{{1}^{2}}}\,+\frac{{{3}^{2}}}{{{1}^{2}}}\,+\frac{{{4}^{2}}}{{{1}^{2}}}\] \[=\frac{4}{1}\,+\frac{9}{1}+\frac{16}{1}\] \[=4+9+16\] = 29                 (iv) \[{{({{3}^{-1}}+{{4}^{-1}}\,+{{5}^{-1}})}^{0}}\]                 \[{{({{3}^{-1}}+{{4}^{-1}}\,+{{5}^{-1}})}^{0}}\]\[={{\left[ \frac{1}{3}+\frac{1}{4}+\frac{1}{5} \right]}^{0}}\] \[={{\left( \frac{20+15+12}{60} \right)}^{0}}\] \[={{\left( \frac{47}{60} \right)}^{0}}\] \[=1\]                 (v) \[{{\left\{ {{\left( \frac{-2}{3} \right)}^{-2}} \right\}}^{2}}\] \[{{\left\{ {{\left( \frac{-2}{3} \right)}^{-2}} \right\}}^{2}}\]\[={{\left( \frac{-2}{3} \right)}^{(-2)\times 2}}\] \[={{\left( \frac{-2}{3} \right)}^{-4}}\] \[=\frac{{{(-2)}^{-4}}}{{{(3)}^{-4}}}\] \[=\frac{{{3}^{4}}}{{{(-2)}^{4}}}\] \[=\frac{3\times 3\times 3\times 3}{(-2)\times (-2)\times (-2)\times (-2)}\] \[=\frac{81}{16}\]