Решение.

Применяем свойства степеней .

[tex]\displaystyle \bf 1.\ \ \frac{5^{n+1}+5^{n}}{6^{n+1}}\cdot \frac{2^{n}}{30^{-n}}=\frac{5^{n}\cdot (5+1)}{3^{n+1}\cdot 2^{n+1}}\cdot 2^{n}\cdot 30^{n}=\frac{6\cdot 5^{n}}{3^{n+1}\cdot 2}\cdot \underbrace{\bf 2^{n}\cdot 3^{n}\cdot 5^{n}}_{30^{n}}=\\\\\\=\frac{5^{n}}{3^{n}}\cdot 2^{n}\cdot 3^{n}\cdot 5^{n}=2^{n}\cdt (5^{n})^2=2^{n}\cdot 25^{n}=50^{n}[/tex]

[tex]\displaystyle \bf \Big(\frac{x^{-2}+1}{1-x^{-2}}\Big)^{-1}=\left(\frac{\dfrac{1}{x^2}+1}{1-\dfrac{1}{x^2}}\right )^{-1}=\left(\frac{(1+x^2)\cdot x^2}{x^2\cdot (x^2-1)}\right)^{-1}=\Big(\frac{1+x^2}{x^2-1}\Big)^{-1}=\frac{x^2-1}{x^2+1}[/tex]