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Z = \(\sqrt{\frac{2^{30 }+ 2^{29}}{2^{29}+{2^{28}}}}\)



\(X= \frac{[(0,2525 ... - \frac{2}{3}): (\frac{16}{33} + 0,343434 ...))]^{-1}}{1,2+\frac{4}{5}}\)



\(W = \frac{1}{\sqrt{5}+\sqrt{8}} + \frac{1}{\sqrt{8}+ \sqrt{11}}+ \frac{1}{\sqrt{11}+\sqrt{14}}+ \frac{1}{\sqrt{14}+\sqrt{17}}+\frac{1}{\sqrt{17}+\sqrt{20}}\)



Y = \((2 + \sqrt{5})^{99} . (2 - \sqrt{5})^{99}\)

\(Z = \sqrt{\frac{2^{30 }+ 2^{29}}{2^{29}+{2^{28}}}}=\sqrt{\frac{2^{28}(2^{2}+ 2^{1})}{2^{28}(2^{1}+2^{0})}}=\sqrt{\frac{4+2}{2+1}}=\sqrt{2}\)



\(X=\frac{(2 + \sqrt{5})^{99}\cdot \left(\frac{16}{33} + 0,343434 ...\right)^{-1}}{1,2 + \frac{4}{5}}=\frac{(2 + \sqrt{5})^{99}\cdot \left(\frac{16}{33} + \frac{34}{99}\right)^{-1}}{\frac{12}{10} + \frac{4}{5}}=\frac{(2 + \sqrt{5})^{99}\cdot \left(\frac{82}{99}\right)^{-1}}{\frac{20}{10}}=\frac{(2 + \sqrt{5})^{99}\cdot \left(\frac{99}{82}\right)}{2}=\frac{(2 + \sqrt{5})^{99}\cdot 99}{164}\)



\(W = \frac{1}{\sqrt{5}+\sqrt{8}} + \frac{1}{\sqrt{8}+ \sqrt{11}}+ \frac{1}{\sqrt{11}+\sqrt{14}}+ \frac{1}{\sqrt{14}+\sqrt{17}}+\frac{1}{\sqrt{17}+\sqrt{20}}= \frac{\sqrt{5}-\sqrt{8}}{(\sqrt{5}+\sqrt{8})\cdot (\sqrt{5}-\sqrt{8})} + \frac{\sqrt{8}-\sqrt{11}}{(\sqrt{8}+\sqrt{11})\cdot (\sqrt{8}-\sqrt{11})}+ \dots +\frac{\sqrt{17}-\sqrt{20}}{(\sqrt{17}+\sqrt{20})\cdot (\sqrt{17}-\sqrt{20})}=\)
\(=\frac{\sqrt{5}-\sqrt{8}}{5-8} + \frac{\sqrt{8}-\sqrt{11}}{8-11}+ \dots +\frac{\sqrt{17}-\sqrt{20}}{17-20}= \frac{\sqrt{5}-\sqrt{8}}{-3} + \frac{\sqrt{8}- \sqrt{11}}{-3}+ \frac{\sqrt{11}-\sqrt{14}}{-3}+ \frac{\sqrt{14}-\sqrt{17}}{-3}+\frac{\sqrt{17}-\sqrt{20}}{-3}=\frac{\sqrt{5}-\sqrt{20}}{-3}=\frac{2\sqrt{5}-\sqrt{5}}{3}=\frac{\sqrt{5}}{3}\)



\(Y = (2 + \sqrt{5})^{99} . (2 - \sqrt{5})^{99}=[(2 + \sqrt{5})\cdot (2 - \sqrt{5})]^{99}=[4-5]^{99}=(-1)^99=-1\)