Question

Mark the correct alternative in the following question:

$$\begin{gathered} \mathrm{If}{\left(\frac{5}{3}\right)}^{-5}\times {\left(\frac{5}{3}\right)}^{11}={\left(\frac{5}{3}\right)}^{8x},\mathrm{then}x=? \\ \left(\mathrm{a}\right)-\frac{1}{2}\left(\mathrm{b}\right)-\frac{3}{4}\left(\mathrm{c}\right)\frac{3}{4}\left(\mathrm{d}\right)\frac{4}{3} \end{gathered}$$

Answer 1

Since,

$$\begin{gathered} {\left(\frac{5}{3}\right)}^{-5}\times {\left(\frac{5}{3}\right)}^{11}={\left(\frac{5}{3}\right)}^{8x} \\ \Rightarrow {\left(\frac{5}{3}\right)}^{-5+11}={\left(\frac{5}{3}\right)}^{8x}\left(\mathrm{As},x^m{\times }^n=x^{m+n}\right) \\ \Rightarrow {\left(\frac{5}{3}\right)}^6={\left(\frac{5}{3}\right)}^{8x} \\ \mathrm{Comparing}\mathrm{the}\mathrm{exponent}\mathrm{of}\mathrm{both}\mathrm{the}\mathrm{sides},\mathrm{we}\mathrm{get} \\ 8x=6 \\ \Rightarrow x=\frac{6}{8} \\ \therefore x=\frac{3}{4} \end{gathered}$$

Hence, the correct alternative is option (c).