Áp dụng các quy tắc tính toán lũy thừa, ta có :

a.  $(\frac{3}{7}+\frac{1}{2})^{2}$

=  $(\frac{6}{14}+\frac{7}{14})^{2}$

=  $(\frac{6+7}{14})^{2}$

=  $(\frac{13}{14})^{2}=\frac{169}{196}$

Vậy  $(\frac{3}{7}+\frac{1}{2})^{2}=\frac{169}{196}$

b.  $(\frac{3}{4}-\frac{5}{6})^{2}$

=  $(\frac{9}{12}-\frac{10}{12})^{2}$

=  $(\frac{9-10}{12})^{2}$

=  $(\frac{-1}{12})^{2}=\frac{1}{144}$

Vậy  $(\frac{3}{4}-\frac{5}{6})^{2}=\frac{1}{144}$

c.  $\frac{5^{4}.20^{4}}{25^{5}.4^{5}}$

=  $\frac{5^{4}.(5.4)^{4}}{(5.5)^{5}.4^{5}}$

=  $\frac{5^{4}.5^{4}.4^{4}}{5^{5}.5^{5}.4^{5}}$

=  $\frac{1}{5.5.4}=\frac{1}{100}$

Vậy  $\frac{5^{4}.20^{4}}{25^{5}.4^{5}}=\frac{1}{100}$

d.  $(\frac{-10}{3})^{5}.(\frac{-6}{5})^{4}$ 

=  $(\frac{-2.5}{3})^{5}.(\frac{-2.3}{5})^{4}$

=  $\frac{(-2.5)^{5}.(3.2)^{4}}{3^{5}.5^{4}}$

=  $\frac{-2^{9.5}}{3}=\frac{-2560}{3}$

Vậy $(\frac{-10}{3})^{5}.(\frac{-6}{5})^{4}=\frac{-2560}{3}$