[tex]xy \\\\=\left(\log_{5\sqrt 5} ~125 \right)\left(\log_{2\sqrt 2} ~64 \right)\\\\=\left(\log_{5\sqrt 5} ~5^3 \right)\left(\log_{2\sqrt 2}~2^6 \right)\\\\=\left(3\log_{5\sqrt 5} ~5 \right)\left(6\log_{2\sqrt 2} ~2 \right)\\\\=\left( 18 \cdot \dfrac{\ln5}{\ln\left(5\sqrt 5\right)} \cdot \dfrac{\ln 2}{ \ln \left( 2 \sqrt 2 \right)}\right)\\\\[/tex]
[tex]=\left( 18 \cdot \dfrac{\ln5}{\ln\left(5^{\tfrac 32}\right)} \cdot \dfrac{\ln 2}{ \ln \left( 2^{\tfrac 32} \right)}\right)~~~~~~~~~~~;\left[\sqrt a = a^{\tfrac 12}~~ \right]\\\\=\left( 18 \cdot \dfrac{\ln5}{\tfrac 32\ln5^} \cdot \dfrac{\ln 2}{\tfrac 32 \ln 2 }\right)\\\\\\=18 \cdot \dfrac 23 \cdot \dfrac 23 \\\\=\dfrac{72}9\\\\=8[/tex]
