Tính \({\left( {\sqrt 3 + \sqrt 2 } \right)^5} – {\left( {\sqrt 3 – \sqrt 2 } \right)^5}\).
Áp dụng công thức khai triển \({(a + b)^5} = {a^5} + 5{a^4}b + 10{a^3}{b^2} + 10{a^2}{b^3} + 5a{b^4} + {b^5}\).
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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}\)