Bài 5.2 phần bài tập bổ sung trang 30 SBT toán 8 tập 1

Đề bài

Thực hiện phép cộng:

\(\displaystyle {1 \over {1 - x}} + {1 \over {1 + x}} \)\(\displaystyle + {2 \over {1 + {x^2}}} + {4 \over {1 + {x^4}}} \)\(\displaystyle + {8 \over {1 + {x^8}}} + {{16} \over {1 + {x^{16}}}}\)-

Lời giải chi tiết

\(\displaystyle {1 \over {1 - x}} + {1 \over {1 + x}} \)\(\displaystyle + {2 \over {1 + {x^2}}} \)\(\displaystyle + {4 \over {1 + {x^4}}}\)\(\displaystyle + {8 \over {1 + {x^8}}}\)\(\displaystyle + {{16} \over {1 + {x^{16}}}}\)

\(\displaystyle   = {{1 + x + 1 - x} \over {\left( {1 - x} \right)\left( {1 + x} \right)}} + {2 \over {1 + {x^2}}} \)\(\displaystyle + {4 \over {1 + {x^4}}} + {8 \over {1 + {x^8}}} \)\(\displaystyle + {{16} \over {1 + {x^{16}}}} \)\(\displaystyle = {2 \over {1 - {x^2}}} + {2 \over {1 + {x^2}}} + {4 \over {1 + {x^4}}} \)\(\displaystyle + {8 \over {1 + {x^8}}} + {{16} \over {1 + {x^{16}}}}  \)

\(\displaystyle = {{2 + 2{x^2} + 2 - 2{x^2}} \over {\left( {1 - {x^2}} \right)\left( {1 + {x^2}} \right)}} \)\(\displaystyle + {4 \over {1 + {x^4}}} + {8 \over {1 + {x^8}}} + {{16} \over {1 + {x^{16}}}}  \)

\(\displaystyle = {4 \over {1 - {x^4}}} + {4 \over {1 + {x^4}}} \)\(\displaystyle + {8 \over {1 + {x^8}}} + {{16} \over {1 + {x^{16}}}}  \)

\(\displaystyle   = {{4 + 4{x^4} + 4 - 4{x^4}} \over {\left( {1 - {x^4}} \right)\left( {1 + {x^4}} \right)}} \)\(\displaystyle + {8 \over {1 + {x^8}}} + {{16} \over {1 + {x^{16}}}}  \)

\(\displaystyle   = {8 \over {1 - {x^8}}} + {8 \over {1 + {x^8}}} + {{16} \over {1 + {x^{16}}}}  \)

\(\displaystyle = {{8 + 8{x^8} + 8 - 8{x^8}} \over {\left( {1 - {x^8}} \right)\left( {1 + {x^8}} \right)}} + {{16} \over {1 + {x^{16}}}} \)

\(\displaystyle = {{16} \over {1 - {x^{16}}}} + {{16} \over {1 + {x^{16}}}}  \)

\(\displaystyle = {{16 + 16{x^{16}} + 16 - 16{x^{16}}} \over {\left( {1 - {x^{16}}} \right)\left( {1 + {x^{16}}} \right)}} \)

\(\displaystyle = {{32} \over {1 - {x^{32}}}} \)