a. \(\displaystyle {{1 - 2x} \over {6{x^3}y}} + {{3 + 2y} \over {6{x^3}y}} + {{2x - 4} \over {6{x^3}y}}\)
\(\displaystyle = {{1 - 2x + 3 + 2y + 2x - 4} \over {6{x^3}y}} \)\(\,\displaystyle= {{2y} \over {6{x^3}y}} = {1 \over {3{x^3}}}\)
b. \(\displaystyle {{{x^2} - 2} \over {x{{\left( {x - 1} \right)}^2}}} + {{2 - x} \over {x{{\left( {x - 1} \right)}^2}}}\)
\( \displaystyle= {{{x^2} - 2 + 2 - x} \over {x{{\left( {x - 1} \right)}^2}}}=\dfrac{{{x^2} - x}}{{x{{\left( {x - 1} \right)}^2}}} \)\(\,\displaystyle = {{x\left( {x - 1} \right)} \over {x{{\left( {x - 1} \right)}^2}}} \displaystyle= {1 \over {x - 1}}\)
c. \(\displaystyle{{3x + 1} \over {{x^2} - 3x + 1}} + {{{x^2} - 6x} \over {{x^2} - 3x + 1}}\)
\(\displaystyle = {{3x + 1 + {x^2} - 6x} \over {{x^2} - 3x + 1}} = {{{x^2} - 3x + 1} \over {{x^2} - 3x + 1}} \)\(\,= 1\)
d. \(\displaystyle {{{x^2} + 38x + 4} \over {2{x^2} + 17x + 1}} + {{3{x^2} - 4x - 2} \over {2{x^2} + 17x + 1}}\)
\(\displaystyle = {{{x^2} + 38x + 4 + 3{x^2} - 4x - 2} \over {2{x^2} + 17x + 1}} \)
\(\displaystyle = {{4{x^2} + 34x + 2} \over {2{x^2} + 17x + 1}}\)
\(\displaystyle = {{2\left( {2{x^2} + 17x + 1} \right)} \over {2{x^2} + 17x + 1}} = 2\)
