Find the multiplicative inverse of : 3+4i/4-5i

Dear Student,

Please find below the solution to the asked query :

$$\begin{gathered} \mathrm{Let} \mathrm{z} = \frac{3+4\mathrm{i}}{4-5\mathrm{i}} \\ \mathrm{Now}, \mathrm{multiplicative} \mathrm{inverse} = \frac{1}{\mathrm{z}} = \frac{4-5\mathrm{i}}{3+4\mathrm{i}} \\ \Rightarrow \frac{1}{\mathrm{z}} = \frac{4-5\mathrm{i}}{3+4\mathrm{i}} = \frac{4-5\mathrm{i}}{3+4\mathrm{i}} \times \frac{3-4\mathrm{i}}{3-4\mathrm{i}} = \frac{12-16\mathrm{i}-15\mathrm{i}-20}{{\left(3\right)}^2-{\left(4\mathrm{i}\right)}^2} = \frac{-31\mathrm{i}-8}{9+16} = \frac{-8-31\mathrm{i}}{25} \end{gathered}$$

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