№51. Вычислить: 1) \(\left(\frac{1}{4}\right)^{-\frac{1}{2}}\cdot25^{\frac{1}{2}}-81^{\frac{1}{2}}\cdot125^{-\frac{1}{3}}\); 2) \(216^{-\frac{1}{3}}\cdot\left(\frac{1}{6}\right)^{-2}-5^{-1}\cdot\left(\frac{1}{25}\right)^{-\frac{1}{2}}\); 3) \(49^{-\frac{1}{2}}\cdot\left(\frac{1}{7}\right)^{-2}+2^{-1}\cdot(-2)^{-2}\); 4) \(\left(\frac{1}{4}\right)^{\frac{1}{2}}\cdot16^{\frac{1}{2}}-2^{-1}\cdot\left(\frac{1}{25}\right)^{-\frac{1}{2}}\cdot8^{\frac{1}{3}}\).

1)
\[\left(\frac{1}{4}\right)^{-\frac{1}{2}}=2,\qquad 25^{\frac{1}{2}}=5,\qquad 81^{\frac{1}{2}}=9,\qquad 125^{-\frac{1}{3}}=\frac{1}{5}.\]
\[2\cdot5-9\cdot\frac{1}{5}=10-\frac{9}{5}=\frac{41}{5}.\]

2)
\[216^{-\frac{1}{3}}=\frac{1}{6},\qquad \left(\frac{1}{6}\right)^{-2}=36,\qquad 5^{-1}=\frac{1}{5},\qquad \left(\frac{1}{25}\right)^{-\frac{1}{2}}=5.\]
\[\frac{1}{6}\cdot36-\frac{1}{5}\cdot5=6-1=5.\]

3)
\[49^{-\frac{1}{2}}=\frac{1}{7},\qquad \left(\frac{1}{7}\right)^{-2}=49,\qquad 2^{-1}=\frac{1}{2},\qquad (-2)^{-2}=\frac{1}{4}.\]
\[\frac{1}{7}\cdot49+\frac{1}{2}\cdot\frac{1}{4}=7+\frac{1}{8}=\frac{57}{8}.\]

4)
\[\left(\frac{1}{4}\right)^{\frac{1}{2}}=\frac{1}{2},\qquad 16^{\frac{1}{2}}=4,\qquad \left(\frac{1}{25}\right)^{-\frac{1}{2}}=5,\qquad 8^{\frac{1}{3}}=2.\]
\[\frac{1}{2}\cdot4-\frac{1}{2}\cdot5\cdot2=2-5=-3.\]

Ответ: 1) \(\frac{41}{5}\); 2) 5; 3) \(\frac{57}{8}\); 4) −3.