Bài 1 trang 49 Toán 8 tập 1 - Cánh diều: Thực hiện phép tính...

Thực hiện phép tính:

$a)\dfrac{x}{{xy + {y^2}}} - \dfrac{y}{{{x^2} + xy}}$

$b)\dfrac{{{x^2} + 4}}{{{x^2} - 4}} - \dfrac{x}{{x + 2}} - \dfrac{x}{{2 - x}}$

$c)\dfrac{{{a^2} + ab}}{{b - a}}:\dfrac{{a + b}}{{2{{\rm{a}}^2} - 2{b^2}}}$

$d)\left( {\dfrac{{2{\rm{x}} + 1}}{{2{\rm{x}} - 1}} - \dfrac{{2{\rm{x}} - 1}}{{2{\rm{x}} + 1}}} \right):\dfrac{{4{\rm{x}}}}{{10{\rm{x}} - 5}}$

Vận dụng các quy tắc cộng, trừ, nhân, chia, hằng đẳng thức phân thức đại số để thực hiện phép tính.

$\begin{array}{l}a)\dfrac{x}{{xy + {y^2}}} - \dfrac{y}{{{x^2} + xy}}\\ = \dfrac{x}{{y\left( {x + y} \right)}} - \dfrac{y}{{x\left( {x + y} \right)}}\\ = \dfrac{{{x^2} - {y^2}}}{{xy\left( {x + y} \right)}} = \dfrac{{\left( {x - y} \right)\left( {x + y} \right)}}{{xy\left( {x + y} \right)}} = \dfrac{{x - y}}{{xy}}\end{array}$

$\begin{array}{l}b)\dfrac{{{x^2} + 4}}{{{x^2} - 4}} - \dfrac{x}{{x + 2}} - \dfrac{x}{{2 - x}}\\ = \dfrac{{{x^2} + 4}}{{\left( {x - 2} \right)\left( {x + 2} \right)}} - \dfrac{x}{{x + 2}} + \dfrac{x}{{x - 2}}\\ = \dfrac{{{x^2} + 4 - x\left( {x - 2} \right) + x\left( {x + 2} \right)}}{{\left( {x - 2} \right)\left( {x + 2} \right)}}\\ = \dfrac{{{x^2} + 4 - {x^2} + 2{\rm{x}} + {x^2} + 2{\rm{x}}}}{{\left( {x - 2} \right)\left( {x + 2} \right)}} = \dfrac{{{x^2} + 4{\rm{x}} + 4}}{{\left( {x - 2} \right)\left( {x + 2} \right)}} = \dfrac{{{{\left( {x + 2} \right)}^2}}}{{\left( {x - 2} \right)\left( {x + 2} \right)}} = \dfrac{{x + 2}}{{x - 2}}\end{array}$

$\begin{array}{l}c)\dfrac{{{a^2} + ab}}{{b - a}}:\dfrac{{a + b}}{{2{{\rm{a}}^2} - 2{b^2}}}\\ = \dfrac{{a\left( {a + b} \right)}}{{b - a}}.\dfrac{{2{{\rm{a}}^2} - 2{b^2}}}{{a + b}}\\ = \dfrac{{a\left( {a + b} \right).2.\left( {{a^2} - {b^2}} \right)}}{{ - \left( {a - b} \right).\left( {a + b} \right)}}\\ = \dfrac{{a\left( {a + b} \right).2.\left( {a - b} \right).\left( {a + b} \right)}}{{ - \left( {a - b} \right)\left( {a + b} \right)}} = - 2{\rm{a}}\left( {a + b} \right)\end{array}$

$\begin{array}{l}d)\left( {\dfrac{{2{\rm{x}} + 1}}{{2{\rm{x}} - 1}} - \dfrac{{2{\rm{x}} - 1}}{{2{\rm{x}} + 1}}} \right):\dfrac{{4{\rm{x}}}}{{10{\rm{x}} - 5}}\\ = \dfrac{{{{\left( {2{\rm{x}} + 1} \right)}^2} - {{\left( {2{\rm{x}} - 1} \right)}^2}}}{{\left( {2{\rm{x}} + 1} \right)\left( {2{\rm{x}} - 1} \right)}}.\dfrac{{10x - 5}}{{4{\rm{x}}}}\\ = \dfrac{{\left( {2{\rm{x}} + 1 + 2{\rm{x}} - 1} \right)\left( {2{\rm{x}} + 1 - 2{\rm{x}} + 1} \right)}}{{\left( {2{\rm{x}} + 1} \right)\left( {2{\rm{x}} - 1} \right)}}.\dfrac{{5.\left( {2{\rm{x}} - 1} \right)}}{{4{\rm{x}}}}\\ = \dfrac{{4{\rm{x}}.2.5\left( {2{\rm{x}} - 1} \right)}}{{\left( {2{\rm{x}} + 1} \right)\left( {2{\rm{x}} - 1} \right).4{\rm{x}}}} = \dfrac{{10}}{{2{\rm{x}} + 1}}\end{array}$