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\(x=\sqrt[3]{9+4\sqrt{5}}+\sqrt[3]{9-4\sqrt{5}}\\ \Leftrightarrow x^3=9+4\sqrt{5}+9-4\sqrt{5}+3\sqrt[3]{\left(9-4\sqrt{5}\right)\left(9+4\sqrt{5}\right)}\left(\sqrt[3]{9+4\sqrt{5}}+\sqrt[3]{9-4\sqrt{5}}\right)\\ \Leftrightarrow x^3=18+3x\sqrt[3]{81-80}=18-3x\\ \Leftrightarrow x^3-3x=18\\ y=\sqrt[3]{3-2\sqrt{2}}+\sqrt[3]{3+2\sqrt{2}}\\ \Leftrightarrow y^3=6+3\sqrt[3]{\left(3-2\sqrt{2}\right)\left(3+2\sqrt{2}\right)}\left(\sqrt[3]{3-2\sqrt{2}}+\sqrt[3]{3+2\sqrt{2}}\right)\\ \Leftrightarrow y^3=6+3y\sqrt[3]{9-8}=6+3y\\ \Leftrightarrow y^3-3y=6\\ \Leftrightarrow P=x^3+y^3-3\left(x+y\right)+1993\\ P=x^3+y^3-3x-3y+1993=18+6+1993=2017\)
Áp dụng: \(\left(a+b\right)^3=a^3+3a^2b+3ab^2+b^3=a^3+b^3+3ab\left(a+b\right)\)
\(x=\sqrt[3]{9+4\sqrt{5}}+\sqrt[3]{9-4\sqrt{5}}\)
\(\Rightarrow x^3=9+4\sqrt{5}+9-4\sqrt{5}+3\sqrt[3]{\left(9+4\sqrt{5}\right)\left(9-4\sqrt{5}\right)}\left(\sqrt[3]{9+4\sqrt{5}}+\sqrt[3]{9-4\sqrt{5}}\right)\)
\(=18+3\sqrt[3]{81-80}.x=18+3x\)
\(y=\sqrt[3]{3-2\sqrt{2}}+\sqrt[3]{3+2\sqrt{2}}\)
\(\Rightarrow y^3=3-2\sqrt{2}+3+2\sqrt{2}+3\sqrt[3]{\left(3-2\sqrt{2}\right)\left(3+2\sqrt{2}\right)}\left(\sqrt[3]{3-2\sqrt{2}}+\sqrt[3]{3+2\sqrt{2}}\right)\)
\(=6+3\sqrt[3]{9-8}y=6+3y\)
\(P=x^3+y^3-3\left(x+y\right)+1993\)
\(=18+3x+6+3y-3x-3y+1993=2017\)
Tá có:
\(\left(x+y\right)^2=3\sqrt{5}-\sqrt{2}=x^2+2xy+y^2.\)
\(\left(x-y\right)^2=3\sqrt{2}-\sqrt{5}=x^2-2xy-y^2\)
\(\Rightarrow\left(x+y\right)^2-\left(x-y\right)^2=4xy=3\sqrt{5}-\sqrt{2}-3\sqrt{2}+\sqrt{5}\)
\(4xy=4\sqrt{5}-4\sqrt{2}\)
\(xy=\sqrt{5}-\sqrt{2}\)
Có : x+y = 6 => x^2+2xy+y^2 = 36
xy = 3^2-2 = 7 <=> 2xy = 14
<=> x^2+y^2 = 22
=> x^4+2x^2y^2+y^4 = 484
<=> x^4+y^4 = 484 - 2x^2y^2 = 484 - 2.(xy)^2 = 484 - 2.7^2 = 386
Xét : 36 x 386 = (x+y).(x^4+y^4) = x^5+y^5+xy.(x^3+y^3) = x^5+y^5+xy.(x+y).(x^2-xy+y^2) = x^5+y^5+7.6.(22-7) = x^5+y^5+630
=> A = x^5+y^5 = 36 x 386 - 630 = 13266
Tk mk nha