Giải bài 8.36 trang 60 SBT toán 10 - Kết nối tri thức

Đề bài

Tính \({\left( {\sqrt 3  + \sqrt 2 } \right)^5} - {\left( {\sqrt 3  - \sqrt 2 } \right)^5}\).

Lời giải chi tiết

 Ta có: \({\left( {\sqrt 3  + \sqrt 2 } \right)^5} - {\left( {\sqrt 3  - \sqrt 2 } \right)^5}\)

\(\begin{array}{l} = {\left( {\sqrt 3 } \right)^5} + 5{\left( {\sqrt 3 } \right)^4}\left( {\sqrt 2 } \right) + 10{\left( {\sqrt 3 } \right)^3}{\left( {\sqrt 2 } \right)^2} + 10{\left( {\sqrt 3 } \right)^2}{\left( {\sqrt 2 } \right)^3} + 5.\left( {\sqrt 3 } \right){\left( {\sqrt 2 } \right)^4} + {\left( {\sqrt 2 } \right)^5}\\ - \left[ {{{\left( {\sqrt 3 } \right)}^5} + 5{{\left( {\sqrt 3 } \right)}^4}\left( { - \sqrt 2 } \right) + 10{{\left( {\sqrt 3 } \right)}^3}{{\left( { - \sqrt 2 } \right)}^2} + 10{{\left( {\sqrt 3 } \right)}^2}{{\left( { - \sqrt 2 } \right)}^3} + 5.\left( {\sqrt 3 } \right){{\left( { - \sqrt 2 } \right)}^4} + {{\left( { - \sqrt 2 } \right)}^5}} \right]\\ = {\left( {\sqrt 3 } \right)^5} + 5{\left( {\sqrt 3 } \right)^4}\left( {\sqrt 2 } \right) + 10{\left( {\sqrt 3 } \right)^3}{\left( {\sqrt 2 } \right)^2} + 10{\left( {\sqrt 3 } \right)^2}{\left( {\sqrt 2 } \right)^3} + 5.\left( {\sqrt 3 } \right){\left( {\sqrt 2 } \right)^4} + {\left( {\sqrt 2 } \right)^5}\\ - \left[ {{{\left( {\sqrt 3 } \right)}^5} - 5{{\left( {\sqrt 3 } \right)}^4}\left( {\sqrt 2 } \right) + 10{{\left( {\sqrt 3 } \right)}^3}{{\left( {\sqrt 2 } \right)}^2} - 10{{\left( {\sqrt 3 } \right)}^2}{{\left( {\sqrt 2 } \right)}^3} + 5.\left( {\sqrt 3 } \right){{\left( {\sqrt 2 } \right)}^4} - {{\left( {\sqrt 2 } \right)}^5}} \right]\\ = {\left( {\sqrt 3 } \right)^5} + 5{\left( {\sqrt 3 } \right)^4}\left( {\sqrt 2 } \right) + 10{\left( {\sqrt 3 } \right)^3}{\left( {\sqrt 2 } \right)^2} + 10{\left( {\sqrt 3 } \right)^2}{\left( {\sqrt 2 } \right)^3} + 5.\left( {\sqrt 3 } \right){\left( {\sqrt 2 } \right)^4} + {\left( {\sqrt 2 } \right)^5}\\ - {\left( {\sqrt 3 } \right)^5} + 5{\left( {\sqrt 3 } \right)^4}\left( {\sqrt 2 } \right) - 10{\left( {\sqrt 3 } \right)^3}{\left( {\sqrt 2 } \right)^2} + 10{\left( {\sqrt 3 } \right)^2}{\left( {\sqrt 2 } \right)^3} - 5.\left( {\sqrt 3 } \right){\left( {\sqrt 2 } \right)^4} + {\left( {\sqrt 2 } \right)^5}\\ = 10{\left( {\sqrt 3 } \right)^4}\left( {\sqrt 2 } \right) + 20{\left( {\sqrt 3 } \right)^2}{\left( {\sqrt 2 } \right)^3} + 2{\left( {\sqrt 2 } \right)^5}\\ = 10.9.\sqrt 2  + 20.3.2.\sqrt 2  + 2.4.\sqrt 2 \\ = 218\sqrt 2 \end{array}\)