Question

Find p : ( 52)7*(5)3*5/510= (53)p

Answer 1

Dear student
$$\begin{gathered} \frac{{\left(5^2\right)}^7\times {\left(5^3\right)}^5}{5^{10}}={\left(5^3\right)}^{\mathrm{p}} \\ \Rightarrow \frac{5^{14}\times 5^{15}}{5^{10}}=5^{3\mathrm{p}} \\ \Rightarrow 5^{14+15-10}=5^{3\mathrm{p}}\left[\mathrm{Since}{\mathrm{a}}^{\mathrm{m}}\times {\mathrm{a}}^{\mathrm{n}}={\mathrm{a}}^{\mathrm{m}+\mathrm{n}}\mathrm{and}\frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}}={\mathrm{a}}^{\mathrm{m}-\mathrm{n}}\right] \\ \Rightarrow 5^{19}=5^{3\mathrm{p}} \\ \mathrm{On}\mathrm{comparing},\mathrm{we}\mathrm{get} \\ 19=3\mathrm{p} \\ \Rightarrow \mathrm{p}=\frac{19}{3} \end{gathered}$$
Regards