x³  + y³ - 3(x +y) +2020 nha các cậu

Đặt \(a=\sqrt[3]{9+4\sqrt{5}},b=\sqrt[3]{9-4\sqrt{5}}\)

\(\Rightarrow\hept{\begin{cases}a^3+b^3=18\\ab=1\end{cases};a+b=x}\)

Ta có: \(x=a+b\Leftrightarrow x^3=\left(a+b\right)^3=a^3+b^3+3ab\left(a+b\right)\)\(\Rightarrow x^3=18+3x\Leftrightarrow x^3-3x=18\)(1)

Tương tự: Đặt \(c=\sqrt[3]{3+2\sqrt{2}},d=\sqrt[3]{3-2\sqrt{2}}\)

\(\Rightarrow\hept{\begin{cases}c^3+d^3=6\\cd=1\end{cases};c+d=y}\)

Ta có: \(y=c+d\Leftrightarrow y^3=\left(c+d\right)^3=c^3+d^3+3cd\left(c+d\right)\)\(\Rightarrow y^3=6+3y\)

\(\Leftrightarrow y^3-3y=6\)(2)

Từ (1) và (2) suy ra \(A=x^3-3x+y^3-3y+2020=18+6+2020=2048\)

áp dụng hằng đẳng thức:\(\left(a+b\right)^3=a^3+b^3+3ab\left(a+b\right)\)ta có

\(x=\sqrt[3]{9+4\sqrt{5}}+\sqrt[3]{9-4\sqrt{5}}\)

\(\Rightarrow x^3=9+4\sqrt{5}+9-4\sqrt{5}+3\sqrt[3]{\left(9+4\sqrt{5}\right)\left(9-4\sqrt{5}\right)}.\left(\sqrt[3]{9+4\sqrt{5}}+\sqrt[3]{9-4\sqrt[3]{5}}\right)\)

\(\Rightarrow x^3=18+3.1.x\)

\(\Rightarrow x^3-3x-18=0\)(1)

\(y=\sqrt[3]{3+2\sqrt{2}}+\sqrt[3]{3-2\sqrt{2}}\)

\(\Rightarrow y^3=3+2\sqrt{2}+3-2\sqrt{2}+3\sqrt{\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)}.\left(\sqrt[3]{3+2\sqrt{2}}+\sqrt[3]{3-2\sqrt{2}}\right)\)

\(\Rightarrow y^3=6+3.1.y\Rightarrow y^3-3y-6=0\)(2)

từ (1), (2) ta có:\(A=x^3+y^3-3x-3y+2020=x^3-3x-18+y^3-3y-6+2044=2044\)

Có sai đề k bạn

Ta có với x,y,z >0 thì:\(\dfrac{x^2}{\sqrt{1-x^2}}=\dfrac{x^3}{x\sqrt{1-x^2}}\)
Bất đẳng thức Cô si ta có:
\(x\sqrt{1-x^2}\le\dfrac{x^2+1-x^2}{2}=\dfrac{1}{2}\\ \Rightarrow\dfrac{1}{x\sqrt{1-x^2}}\ge2\\ \Rightarrow\dfrac{x^3}{x\sqrt{1-x^2}}\ge2x^3\Leftrightarrow\dfrac{x^2}{\sqrt{1-x^2}}\ge2x^3\)
Tương tự: \(\dfrac{y^2}{\sqrt{1-y^2}}\ge2y^3;\dfrac{z^2}{\sqrt{1-z^2}}\ge2z^3\)
Từ đó ta có:\(\dfrac{x^2}{\sqrt{1-x^2}}+\dfrac{y^2}{\sqrt{1-y^2}}+\dfrac{z^2}{\sqrt{1-z^2}}\ge2\left(x^3+y^3+z^3\right)=2\left(dpcm\right)\)
 

Kiểm tra lại đề bài đi em, chỗ CMR đó

Đặt \(\sqrt[3]{x^2}=m\Leftrightarrow x^2=m^3;\sqrt[3]{y^2}=n\Leftrightarrow y^2=n^3\)

Thay vào biểu thức:

\(\Leftrightarrow\sqrt{m^3+m^2n}+\sqrt{n^3+mn^2}=a\\ \Leftrightarrow m^3+n^3+mn\left(m+n\right)+2\sqrt{\left(m^3+m^2n\right)\left(n^3+mn^2\right)}=a^2\\ \Leftrightarrow m^3+n^3+mn\left(m+n\right)+2\sqrt{m^2n^2\left(m+n\right)}=a^2\\ \Leftrightarrow m^3+n^3+3mn\left(m+n\right)=a^2\\ \Leftrightarrow\left(m+n\right)^3=a^2\\ \Leftrightarrow m+n=\sqrt[3]{a^2}\\ \Leftrightarrow\sqrt[3]{x^2}+\sqrt[3]{y^2}=\sqrt[3]{a^2}\)

Em chắc chắn là đề bài đúng chứ? Trước khi nhìn kĩ lại?

\(x^3=3+2\sqrt{2}+3-2\sqrt{2}+3\cdot\sqrt[3]{\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)}\left(\sqrt[3]{3+2\sqrt{2}}+\sqrt[3]{3-2\sqrt{2}}\right)\\ \Leftrightarrow x^3=6+3x\sqrt[3]{1}\\ \Leftrightarrow x^3-3x=6\)

\(y^3=17+12\sqrt{2}+17-12\sqrt{2}+3\sqrt[3]{\left(17-12\sqrt{2}\right)\left(17+12\sqrt{2}\right)}\left(\sqrt[3]{17-12\sqrt{2}}+\sqrt[3]{17+12\sqrt{2}}\right)\\ \Leftrightarrow y^3=34+3x\sqrt[3]{1}\\ \Leftrightarrow y^3-3y=34\)

Thay vào P, ta được

\(P=x^3+y^3-3x-3y+1979\\ P=\left(x^3-3x\right)+\left(y^3-3y\right)+1979\\ P=6+34+1979=2019\)

\(x^3=6+3\sqrt[3]{\left(3+2\sqrt[]{2}\right)\left(3-2\sqrt[]{2}\right)}\left(\sqrt[3]{3+2\sqrt[]{2}}+\sqrt[3]{3-2\sqrt[]{2}}\right)\)

\(\Rightarrow x^3=6+3x\)

\(\Rightarrow x^3-3x=6\)

Tương tự:

\(y^3=34+3\sqrt[3]{\left(17+12\sqrt[]{2}\right)\left(17-12\sqrt[]{2}\right)}\left(\sqrt[3]{17+12\sqrt[]{2}}+\sqrt[3]{17-12\sqrt[]{2}}\right)\)

\(\Rightarrow y^3=34+3y\)

\(\Rightarrow y^3-3y=34\)

Do đó:

\(P=\left(x^3-3x\right)+\left(y^3-3y\right)+1979=6+34+1979=...\)

\(x=\sqrt[3]{3+2\sqrt{2}}+\sqrt[3]{3-2\sqrt{2}}\)

\(\Rightarrow x^3=3+2\sqrt{2}+3-2\sqrt{2}+3\sqrt[3]{\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)}\left(\sqrt[3]{3+2\sqrt{2}}+\sqrt[3]{3-2\sqrt{2}}\right)\)

\(=6+3\sqrt[3]{9-8}.x=6+3x\)

\(\Rightarrow x^3-3x=6\)

\(y=\sqrt[3]{17+12\sqrt{2}}+\sqrt[3]{17-12\sqrt{2}}\)

\(\Rightarrow y^3=17+12\sqrt{2}+17-12\sqrt{2}+3\sqrt[3]{\left(17+12\sqrt{2}\right)\left(17-12\sqrt{2}\right)}\left(\sqrt[3]{17+12\sqrt{2}}+\sqrt[3]{17-12\sqrt{2}}\right)\)

\(=34+3\sqrt[3]{289-288}.y=34+3y\)

\(\Rightarrow y^3-3y=34\)

\(P=x^3+y^3-3\left(x+y\right)+2009=\left(x^3-3x\right)+\left(y^3-3y\right)+2009\)

\(=6+34+2009=2049\)

Có \(x^3=3+2\sqrt{2}-3\sqrt[3]{\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)}\left(\sqrt[3]{3+2\sqrt{2}}-\sqrt[3]{3-2\sqrt{2}}\right)-\left(3-2\sqrt{2}\right)\)

\(\Leftrightarrow x^3=4\sqrt{2}-3x\) \(\Leftrightarrow x^3+3x=4\sqrt{2}\) (1)

Có \(y^3=17+12\sqrt{2}-3\sqrt[3]{\left(17+12\sqrt{2}\right)\left(17-12\sqrt{2}\right)}\left(\sqrt[3]{17+12\sqrt{2}}-\sqrt[3]{17-12\sqrt{2}}\right)-\left(17-12\sqrt{2}\right)\)

\(\Leftrightarrow y^3=24\sqrt{2}-3y\) \(\Leftrightarrow y^3+3y=24\sqrt{2}\) (2)

Từ (1) (2)\(\Rightarrow x^3+3x-y^3-3y=-20\sqrt{2}\)

Có \(M=\left(x-y\right)^3+3\left(x-y\right)\left(xy+1\right)=\left(x-y\right)\left[\left(x-y\right)^2+3\left(xy+1\right)\right]\)

\(=\left(x-y\right)\left(x^2+xy+y^2+3\right)=x^3-y^3+3\left(x-y\right)=-20\sqrt{2}\)

Vậy \(M=-20\sqrt{2}\)

theo bài ra

\(x=\sqrt[3]{3+2\sqrt{2}}-\sqrt[3]{3-2\sqrt{2}}\)

\(=>x^3=\left(\sqrt[3]{3+2\sqrt{2}}-\sqrt[3]{3-2\sqrt{2}}\right)^3\)

\(x^3=4\sqrt{2}-3\left[\left(\sqrt[3]{3+2\sqrt{2}}\right)\left(\sqrt[3]{3-2\sqrt{2}}\right)\right]\left[\sqrt[3]{3+2\sqrt{2}}-\sqrt[3]{3-2\sqrt{2}}\right]\)

\(x^3=4\sqrt{2}-3\left[\sqrt[3]{\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)}\right].x\)

\(x^3=4\sqrt{2}-3.\left[\sqrt[3]{9-\left(2\sqrt{2}\right)^2}\right]x\)

\(x^3=4\sqrt{2}-3.1x\)

\(x^3=4\sqrt{2}-3x\)

\(< =>x^3+3x-4\sqrt{2}=0\)

rồi làm y tương tự rồi thế vào M là ra