(20^ {\frac{3}{4}} * 5^{1 \frac{1}{4}} ) : 2^{- \frac{1}{2}}=(20^{\frac{3}{4}}*5^{\frac{5}{4}}):2^{-\frac{1}{2}}=(20^{\frac{3}{4}}*5^{\frac{3}{4}}*5^{\frac{2}{4}}):(\frac{1}{2})^{\frac{1}{2}}=
=[(20*5)^{\frac{3}{4}}*5^{\frac{1}{2}}]:\sqrt{\frac{1}{2}}=[(10^2)^{\frac{3}{4}}*5^{\frac{1}{2}}]:\frac{1}{\sqrt2}=(10^{\frac{3}{2}}*5^{\frac{1}{2}})*\frac{\sqrt2}{1}=
=10\sqrt{10}*\sqrt5*\sqrt2=10\sqrt{100}=10*10=100