Question

Evaluate ${\left\{{\left(\frac{4}{3}\right)}^{-1}-{\left(\frac{1}{4}\right)}^{-1}\right\}}^{-1}.$

Answer 1

$$\begin{gathered} {\left\{{\left(\frac{4}{3}\right)}^{-1}-{\left(\frac{1}{4}\right)}^{-1}\right\}}^{-1}={\left\{{\left(\frac{3}{4}\right)}^1-{\left(\frac{4}{1}\right)}^1\right\}}^{-1}={\left\{\left(\frac{3}{4}\right)-\left(\frac{4}{1}\right)\right\}}^{-1} \\ \mathrm{The}\mathrm{L}.\mathrm{C}.\mathrm{M}.\mathrm{of}4\mathrm{and}1\mathrm{is}4. \\ \therefore {\left\{\left(\frac{3\times 1}{4\times 1}\right)-\left(\frac{4\times 4}{1\times 4}\right)\right\}}^{-1} \\ ={\left\{\frac{3}{4}-\frac{16}{4}\right\}}^{-1}={\left\{\frac{3-16}{4}\right\}}^{-1}={\left\{\frac{-13}{4}\right\}}^{-1}={\left\{\frac{4}{-13}\right\}}^1=\frac{4}{-13} \\ =\frac{4\times -1}{-13\times -1}=\frac{-4}{13} \end{gathered}$$