\( 0 \)

\( \dfrac{2}{x}+\dfrac{2}{xy}-\dfrac{2(y+1)}{xy}= \dfrac{\;\;\;\;2^{\;\;(y}}{x}+\dfrac{\;\;\;\;2^{\;\;(1}}{xy}-\dfrac{2(y+1)^{\;\;(1}}{xy}= \dfrac{2y+2-2(y+1)}{xy}= \)

\(\dfrac{2y+2-2y-2}{xy}=\dfrac{0}{xy}=0 \)

\( 0 \)

\( \dfrac{5}{a}-\dfrac{5}{ab}-\dfrac{5(b-1)}{ab}= \dfrac{\;\;\;\;5^{\;\;(b}}{a}-\dfrac{\;\;\;\;5^{\;\;(1}}{ab}-\dfrac{5(b-1)^{\;\;(1}}{ab}= \dfrac{5b-5-5(b-1)}{ab}= \)

\(\dfrac{5b-5-5b+5}{ab}=\dfrac{0}{ab}=0 \)

\( 0 \)

\( \dfrac{2a}{x}-\dfrac{18a}{xy}-\dfrac{2ay-18a}{xy} = \dfrac{\;\;\;\;2a^{\;\;(y}}{x}-\dfrac{\;\;\;\;18a^{\;\;(1}}{xy}-\dfrac{2ay-18a^{\;\;(1}}{xy}= \)

\( \dfrac{2ay-18a-(2ay-18a)}{xy}= \dfrac{2ay-18a-2ay+18a}{xy}=\dfrac{0}{xy}=0 \)

\( 1 \)

\(1)\;\;\;\; \dfrac{1}{a+b}+\dfrac{1}{a-b}= \dfrac{\;\;\;\;\;\;1^{\;\;(a-b}}{a+b}+\dfrac{\;\;\;\;\;\;\;1^{\;\;(a+b}}{a-b}= \dfrac{a-b+a+b}{(a-b)(a+b)} = \)

\(= \dfrac{2a}{(a-b)(a+b)} \)


\( 2) \;\;\;\; \dfrac{2a}{(a-b)(a+b)}\cdot \dfrac{(a+b)(a-b)}{2a}=1 \)

\( -1 \)

\(1)\;\;\;\; \dfrac{1}{a+b}-\dfrac{1}{a-b}= \dfrac{\;\;\;\;\;\;1^{\;\;(a-b}}{a+b}-\dfrac{\;\;\;\;\;\;\;1^{\;\;(a+b}}{a-b}= \dfrac{a-b-a-b}{(a-b)(a+b)} = \)

\(= \dfrac{-2b}{(a-b)(a+b)} \)


\( 2) \;\;\;\; \dfrac{-2b}{(a-b)(a+b)}\cdot \dfrac{(a+b)(a-b)}{2b}=-1 \)

\( 1 \)

\(1)\;\;\;\; \dfrac{2}{a-b}-\dfrac{1}{a+b}= \dfrac{\;\;\;\;\;\;2^{\;\;(a+b}}{a+b}-\dfrac{\;\;\;\;\;\;\;1^{\;\;(a-b}}{a-b}= \dfrac{2a+2b-1\cdot(a-b)}{(a-b)(a+b)} = \)

\(= \dfrac{2a+2b-a+b}{(a-b)(a+b)}=\dfrac{a+3b}{(a-b)(a+b)} \)


\( 2) \;\;\;\; \dfrac{a+3b}{(a-b)(a+b)}\cdot \dfrac{(a+b)(a-b)}{a+3b}=1 \)

\( 1 \)

\( 1)\;\;\; 1-\dfrac{1}{a}=\dfrac{1}{1}-\dfrac{1}{a}=\dfrac{\;\;\;\;1^{\;\;(a}}{1}-\dfrac{\;\;\;\;1^{\;\;(1}}{a}= \dfrac{a-1}{a} \)


\( 2) \;\;\;\; \dfrac{a-1}{a}\cdot \dfrac{a}{a-1}=1 \)

\( 1 \)

\( 1)\;\;\; \dfrac{1}{b}-\dfrac{1}{a}=\dfrac{\;\;\;\;1^{\;\;(a}}{b}-\dfrac{\;\;\;\;1^{\;\;(b}}{a}=\dfrac{a-b}{ba} \)


\( 2)\;\;\; \dfrac{a-b}{ba}\cdot \dfrac{ab}{a-b}=1 \)