Câu 58 trang 39 Sách bài tập (SBT) Toán 8 tập 1

Thực hiện các phép tính :

a. \(\left( {{9 \over {{x^3} - 9x}} + {1 \over {x + 3}}} \right):\left( {{{x - 3} \over {{x^2} + 3x}} - {x \over {3x + 9}}} \right)\)

b. \(\left( {{2 \over {x - 2}} - {2 \over {x + 2}}} \right).{{{x^2} + 4x + 4} \over 8}\)

c. \(\left( {{{3x} \over {1 - 3x}} + {{2x} \over {3x + 1}}} \right):{{6{x^2} + 10x} \over {1 - 6x + 9{x^2}}}\)

d. \(\left( {{x \over {{x^2} - 25}} - {{x - 5} \over {{x^2} + 5x}}} \right):{{2x - 5} \over {{x^2} + 5x}} + {x \over {5 - x}}\)

e. \(\left( {{{{x^2} + xy} \over {{x^3} + {x^2}y + x{y^2} + {y^3}}} + {y \over {{x^2} + {y^2}}}} \right):\left( {{1 \over {x - y}} - {{2xy} \over {{x^3} - {x^2}y + x{y^2} - {y^3}}}} \right)\)

Giải:

a. \(\left( {{9 \over {{x^3} - 9x}} + {1 \over {x + 3}}} \right):\left( {{{x - 3} \over {{x^2} + 3x}} - {x \over {3x + 9}}} \right)\)

\(\eqalign{  &  = \left[ {{9 \over {x\left( {x + 3} \right)\left( {x - 3} \right)}} + {1 \over {x + 3}}} \right]:\left[ {{{x - 3} \over {x\left( {x + 3} \right)}} - {x \over {3\left( {x + 3} \right)}}} \right]  \cr  &  = {{9 + x\left( {x - 3} \right)} \over {x\left( {x + 3} \right)\left( {x - 3} \right)}}:{{3\left( {x - 3} \right) - {x^2}} \over {3x\left( {x + 3} \right)}} = {{{x^2} - 3x + 9} \over {x\left( {x + 3} \right)\left( {x - 3} \right)}}.{{3x\left( {x + 3} \right)} \over {3x - 9 - {x^2}}}  \cr  &  = {{3\left( {{x^2} - 3x + 9} \right)} \over {\left( {3 - x} \right)\left( {{x^2} - 3x + 9} \right)}} = {3 \over {3 - x}} \cr} \)

b. \(\left( {{2 \over {x - 2}} - {2 \over {x + 2}}} \right).{{{x^2} + 4x + 4} \over 8}\)\( = {{2\left( {x + 2} \right) - 2\left( {x - 2} \right)} \over {\left( {x - 2} \right)\left( {x + 2} \right)}}.{{{{\left( {x + 2} \right)}^2}} \over 8}\)

\( = {{2x + 4 - 2x + 4} \over {\left( {x - 2} \right)\left( {x + 2} \right)}}.{{{{\left( {x + 2} \right)}^2}} \over 8} = {8 \over {\left( {x - 2} \right)\left( {x + 2} \right)}}.{{{{\left( {x + 2} \right)}^2}} \over 8} = {{x + 2} \over {x - 2}}\)

c. \(\left( {{{3x} \over {1 - 3x}} + {{2x} \over {3x + 1}}} \right):{{6{x^2} + 10x} \over {1 - 6x + 9{x^2}}}\)\( = {{3x\left( {3x + 1} \right) + 2x\left( {1 - 3x} \right)} \over {\left( {1 - 3x} \right)\left( {1 + 3x} \right)}}:{{2x\left( {3x + 5} \right)} \over {{{\left( {1 - 3x} \right)}^2}}}\)

\(\eqalign{  &  = {{9{x^2} + 3x + 2x - 6{x^2}} \over {\left( {1 - 3x} \right)\left( {1 + 3x} \right)}}.{{{{\left( {1 - 3x} \right)}^2}} \over {2x\left( {3x + 5} \right)}} = {{x\left( {3x + 5} \right)} \over {\left( {1 - 3x} \right)\left( {1 + 3x} \right)}}.{{{{\left( {1 - 3x} \right)}^2}} \over {2x\left( {3x + 5} \right)}}  \cr  &  = {{1 - 3x} \over {2\left( {1 + 3x} \right)}} \cr} \)

d. \(\left( {{x \over {{x^2} - 25}} - {{x - 5} \over {{x^2} + 5x}}} \right):{{2x - 5} \over {{x^2} + 5x}} + {x \over {5 - x}}\)

\(\eqalign{ &  = \left[ {{x \over {\left( {x + 5} \right)\left( {x - 5} \right)}} - {{x - 5} \over {x\left( {x + 5} \right)}}} \right]:{{2x - 5} \over {x\left( {x + 5} \right)}} + {x \over {5 - x}}  \cr  &  = {{{x^2} - {{\left( {x - 5} \right)}^2}} \over {x\left( {x + 5} \right)\left( {x - 5} \right)}}.{{x\left( {x + 5} \right)} \over {2x - 5}} + {x \over {5 - x}}  \cr  &  = {{{x^2} - {x^2} + 10x - 25} \over {\left( {x - 5} \right)\left( {2x - 5} \right)}} + {x \over {5 - x}} = {{5\left( {2x - 5} \right)} \over {\left( {x - 5} \right)\left( {2x - 5} \right)}} - {x \over {x - 5}}  \cr  &  = {5 \over {x - 5}} - {x \over {x - 5}} = {{5 - x} \over {x - 5}} = {{ - \left( {x - 5} \right)} \over {x - 5}} =  - 1 \cr} \)

e. \(\left( {{{{x^2} + xy} \over {{x^3} + {x^2}y + x{y^2} + {y^3}}} + {y \over {{x^2} + {y^2}}}} \right):\left( {{1 \over {x - y}} - {{2xy} \over {{x^3} - {x^2}y + x{y^2} - {y^3}}}} \right)\)

\(\eqalign{  &  = \left[ {{{{x^2} + xy} \over {\left( {{x^2} + {y^2}} \right)\left( {x + y} \right)}} + {y \over {{x^2} + {y^2}}}} \right]:\left[ {{1 \over {x - y}} - {{2xy} \over {\left( {{x^2} + {y^2}} \right)\left( {x - y} \right)}}} \right]  \cr  &  = {{{x^2} + xy + y\left( {x + y} \right)} \over {\left( {{x^2} + {y^2}} \right)\left( {x + y} \right)}}:{{{x^2} + {y^2} - 2xy} \over {\left( {{x^2} + {y^2}} \right)\left( {x - y} \right)}}  \cr  &  = {{{x^2} + xy + xy + {y^2}} \over {\left( {{x^2} + {y^2}} \right)\left( {x + y} \right)}}.{{\left( {{x^2} + {y^2}} \right)\left( {x - y} \right)} \over {{{\left( {x - y} \right)}^2}}}  \cr  &  = {{{{\left( {x + y} \right)}^2}} \over {\left( {{x^2} + {y^2}} \right)\left( {x + y} \right)}}.{{\left( {{x^2} + {y^2}} \right)\left( {x - y} \right)} \over {{{\left( {x - y} \right)}^2}}} = {{x + y} \over {x - y}} \cr} \)