Tính giá trị của các biểu thức

a) ${\left(\frac{27}{8}\right)}^{\frac{5}{6}}.{\left(\frac{4^{\frac{3}{2}}}{3^3}\right)}^{\frac{1}{2}}$ ;                                       

b) $\log \sqrt{5}+\log \sqrt{2}$ ;

c) ${\left(\frac{16}{81}\right)}^{-\frac{3}{4}}+{\log }_5\frac{9}{4}+{\log }_5\frac{4}{9}$ ;                   

d) ${\log }_{2}7.{\log }_{3}16.{\log }_{9}3.{\log }_{7}9$ .

a) ${\left(\frac{3^3}{2^3}\right)}^{\frac{5}{6}}.\left(\frac{4^{\frac{3}{4}}}{3^{\frac{3}{2}}}\right)={\left(\frac{3}{2}\right)}^{3.\frac{5}{6}}.\left(\frac{4^{\frac{3}{4}}}{3^{\frac{3}{2}}}\right)$

$={\left(\frac{3}{2}\right)}^{\frac{5}{2}}.\left(\frac{2^{2.\frac{3}{4}}}{3^{\frac{3}{2}}}\right)$$=\frac{3}{2}$

b)   $\log \sqrt{5}+\log \sqrt{2}=\log \sqrt{5}.\sqrt{2}$

$=\log \sqrt{10}$$=\log {10}^{\frac{1}{2}}=\frac{1}{2}$

c) ${\left(\frac{16}{81}\right)}^{-\frac{3}{4}}+{\log }_5\frac{9}{4}+{\log }_5\frac{4}{9}$

$={\left(\frac{2}{3}\right)}^{4\cdot \left(-\frac{3}{4}\right)}+{\log }_5\left(\frac{9}{4}\cdot \frac{4}{9}\right)$

$={\left(\frac{2}{3}\right)}^{-3}+0=\frac{27}{8}$

d) ${\log }_{2}7.{\log }_{3}16.{\log }_{9}3.{\log }_{7}9$

= ${\log }_{2}7.{\log }_3{2}^4.{\log }_{3^2}3.{\log }_7{3}^2$

$=4.{\log }_{2}7.{\log }_{7}3.{\log }_{3}2=4$