(25)^7.5 × (5)^2.5 ÷ (125)^1.5 = 5^?

Solution (By Examveda Team)

$$\eqalign{ & {\text{Let}}\,{\left( {25} \right)^{7.5}} \times {\left( 5 \right)^{2.5}} \div {\left( {125} \right)^{1.5}} = {5^x} \cr & {\text{Then}},\,\frac{{{{\left( {{5^2}} \right)}^{7.5}} \times {{\left( 5 \right)}^{2.5}}}}{{{{\left( {{5^3}} \right)}^{1.5}}}} = {5^x} \cr & \Rightarrow \frac{{{5^{\left( {2 \times 7.5} \right)}} \times {5^{2.5}}}}{{{5^{\left( {3 \times 1.5} \right)}}}} = {5^x} \cr & \Rightarrow \frac{{{5^{15}} \times {5^{2.5}}}}{{{5^{4.5}}}} = {5^x} \cr & \Rightarrow {5^x} = {5^{\left( {15 + 2.5 - 4.5} \right)}} \cr & \Rightarrow {5^x} = {5^{13}} \cr & \therefore x = 13 \cr} $$