Giải bài 15 trang 27 sách bài tập toán 8 - Chân trời sáng tạo

Đề bài

Tính:

a) \(\left( {a + 1 + \frac{{1 - 2{a^2}}}{{a - 1}}} \right):\left( {1 - \frac{1}{{1 - a}}} \right)\);

b) \(\left( {\frac{a}{{{b^2}}} - \frac{1}{a}} \right):\left( {\frac{1}{b} + \frac{1}{a}} \right)\);

c) \(\left( {a - \frac{{4ab}}{{a + b}} + b} \right).\left( {a + \frac{{4ab}}{{a - b}} - b} \right)\);

d) \(ab + \frac{{ab}}{{a + b}}\left( {\frac{{a + b}}{{a - b}} - a - b} \right)\).

Lời giải chi tiết

a) \(\left( {a + 1 + \frac{{1 - 2{a^2}}}{{a - 1}}} \right):\left( {1 - \frac{1}{{1 - a}}} \right) = \frac{{\left( {a + 1} \right)\left( {a - 1} \right) + 1 - 2{a^2}}}{{a - 1}}:\frac{{1 - a - 1}}{{1 - a}}\)

\( = \frac{{{a^2} - 1 + 1 - 2{a^2}}}{{a - 1}}.\frac{{a - 1}}{a} = \frac{{ - {a^2}\left( {a - 1} \right)}}{{a\left( {a - 1} \right)}} =  - a\)

b) \(\left( {\frac{a}{{{b^2}}} - \frac{1}{a}} \right):\left( {\frac{1}{b} + \frac{1}{a}} \right) = \frac{{{a^2} - {b^2}}}{{a{b^2}}}:\frac{{a + b}}{{ab}} = \frac{{\left( {a - b} \right)\left( {a + b} \right)ab}}{{a{b^2}\left( {a + b} \right)}} = \frac{{a - b}}{b}\)

c) \(\left( {a - \frac{{4ab}}{{a + b}} + b} \right).\left( {a + \frac{{4ab}}{{a - b}} - b} \right) = \frac{{\left( {a + b} \right)\left( {a + b} \right) - 4ab}}{{a + b}}.\frac{{\left( {a - b} \right)\left( {a - b} \right) + 4ab}}{{a - b}}\)

\( = \frac{{{a^2} + 2ab + {b^2} - 4ab}}{{a + b}}.\frac{{{a^2} - 2ab + {b^2} + 4ab}}{{a - b}} = \frac{{{a^2} - 2ab + {b^2}}}{{a + b}}.\frac{{{a^2} + 2ab + {b^2}}}{{a - b}}\)

\( = \frac{{{{\left( {a - b} \right)}^2}{{\left( {a + b} \right)}^2}}}{{\left( {a + b} \right)\left( {a - b} \right)}} = \left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}\)

d) \(ab + \frac{{ab}}{{a + b}}\left( {\frac{{a + b}}{{a - b}} - a - b} \right) = ab + \frac{{ab}}{{a + b}}.\frac{{a + b - \left( {a - b} \right)\left( {a + b} \right)}}{{a - b}}\)

\( = ab + \frac{{ab}}{{a + b}}.\frac{{\left( {a + b} \right)\left( {1 - a + b} \right)}}{{a - b}} = ab + \frac{{ab\left( {1 - a + b} \right)}}{{a - b}} = \frac{{ab\left( {a - b} \right) + ab - {a^2}b + a{b^2}}}{{a - b}}\)

\( = \frac{{{a^2}b - a{b^2} + ab - {a^2}b + a{b^2}}}{{a - b}} = \frac{{\left( {{a^2}b - {a^2}b} \right) + \left( {a{b^2} - a{b^2}} \right) + ab}}{{a - b}} = \frac{{ab}}{{a - b}}\)