question_answer

Simplify the following:

(a) \[{{({{6}^{-1}}-{{8}^{-1}})}^{-1}}+{{({{2}^{-1}}-{{3}^{-1}})}^{-1}}\]

(b) \[{{\left\{ {{6}^{-1}}+{{\left( \frac{3}{2} \right)}^{-1}} \right\}}^{-1}}\]

 

Answer:

(a) \[\left( {{6}^{-1}}-{{8}^{-1}} \right)+\left( {{2}^{-1}}+{{3}^{-1}} \right)\]

\[={{\left( \frac{1}{6}-\frac{1}{8} \right)}^{-1}}+{{\left( \frac{1}{2}-\frac{1}{3} \right)}^{-1}}\]

\[={{\left( \frac{4-3}{24} \right)}^{-1}}+{{\left( \frac{3-2}{6} \right)}^{-1}}\]

\[={{\left( \frac{1}{24} \right)}^{-1}}+{{\left( \frac{1}{6} \right)}^{-1}}\]

= 24 + 6 = 30

(b) \[{{\left\{ {{6}^{-1}}+\left( {{\frac{3}{2}}^{-1}} \right) \right\}}^{-1}}={{\left( \frac{1}{6}+\frac{2}{3} \right)}^{-1}}\]

\[={{\left( \frac{1+4}{6} \right)}^{-1}}=\left( \frac{5}{6} \right)\]

\[=\frac{6}{5}\]