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18 tháng 7 2016
Bài 32:
a) P= \(\frac{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)
= \(\frac{\left(\sqrt{2}+\sqrt{3}+\sqrt{4}\right)+\left(\sqrt{4}+\sqrt{6}+\sqrt{8}\right)}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)
= \(\frac{\left(\sqrt{2}+\sqrt{3}+\sqrt{4}\right)+\sqrt{2}\left(\sqrt{2}+\sqrt{3}+\sqrt{4}\right)}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)
= \(\frac{\left(\sqrt{2}+\sqrt{3}+\sqrt{4}\right)\left(1+\sqrt{2}\right)}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)
= \(1+\sqrt{2}\)
b) Có: \(x^2-2y^2=xy\)
\(\Leftrightarrow x^2-y^2-y^2-xy=0\)
\(\Leftrightarrow\left(x-y\right)\left(x+y\right)-y\left(y+x\right)\)
\(\Leftrightarrow\left(x+y\right)\left(x-y-y\right)=0\)
\(\Leftrightarrow\left(x+y\right)\left(x-2y\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x+y=0\\x-2y=0\end{cases}\Leftrightarrow\orbr{\begin{cases}x=-y\\x=2y\end{cases}}}\)
Thay x=-y ta có: Q=\(\frac{-y-y}{-y+y}\)=\(\frac{-2y}{0}\)(loại )
Thay x=2y ta có : Q=\(\frac{2y-y}{2y+y}=\frac{y}{3y}=\frac{1}{3}\)
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\(y+z=-x\)
\(\left(y+z\right)^5=-x^5\)
\(y^5+5y^4z+10y^3z^2+10y^2z^3+5yz^4+z^5+x^5=0\)
\(x^5+y^5+z^5+5yz\left(y^3+2y^2z+2yz^2+z^3\right)=0\)
\(x^5+y^5+z^5+5yz\left(\left(y+z\right)\left(y^2-yz+z^2\right)+2yz\left(y+z\right)\right)=0\)
\(x^5+y^5+z^5+5yz\left(y+z\right)\left(y^2+yz+z^2\right)=0\)
\(2\left(x^5+y^5+z^5\right)-5xyz\left(\left(y^2+2yz+z^2\right)+y^2+z^2\right)=0\)
\(2\left(x^5+y^5+z^5\right)=5xyz\left(x^2+y^2+z^2\right)\)
Ta có: \(y+z=-x\)
\(\left(y+z\right)^5=-x^5\)
\(y^5+5y^4z+10y^3z^2+10y^2z^3+5yz^4+z^5+x^5=0\)
\(x^5+y^5+z^5+5yz\left(y^3+2y^2z+2yz^2+z^3\right)=0\)
\(x^5+y^5+z^5+5yz\left(\left(y+z\right)\left(y^2-yz+z^2\right)+2yz\left(y+z\right)\right)=0\)
\(x^5+y^5+z^5+5yz\left(y+z\right)\left(y^2+yz+z^2\right)=0\)
\(2\left(x^5+y^5+z^5\right)-5xyz\left(\left(y^2+2yz+z^2\right)+y^2+z^2\right)=0\)
\(2\left(x^5+y^5+z^5\right)=5xyz\left(x^2+y^2+z^2\right)\)
Ta có: \(x+y+z=0\Rightarrow x+y=-z\Rightarrow\left(x+y\right)^3=\left(-z\right)^3\Rightarrow x^3+y^3+3xy\left(x+y\right)=-z^3\Rightarrow x^3+y^3+z^3=-3xy\left(x+y\right)=-3xy.\left(-z\right)=3xyz\Rightarrow\left(x^2+y^2+z^2\right)\left(x^3+y^3+z^3\right)=3xyz\left(x^2+y^2+z^2\right)\)\(\Leftrightarrow x^5+y^5+z^5+x^3\left(y^2+z^2\right)+y^3\left(z^2+x^2\right)+z^3\left(x^2+y^2\right)=3xyz\left(x^2+y^2+z^2\right)\Leftrightarrow x^5+y^5+z^5+x^3\left[\left(y+z\right)^2-2yz\right]+y^3\left[\left(z+x\right)^2-2zx\right]+z^3\left[\left(x+y\right)^2-2xy\right]=3xyz\left(x^2+y^2+z^2\right)\)\(\Leftrightarrow x^5+y^5+z^5+x^3\left[x^2-2yz\right]+y^3\left[y^2-2zx\right]+z^3\left[z^2-2xy\right]=3xyz\left(x^2+y^2+z^2\right)\Leftrightarrow2\left(x^5+y^5+z^5\right)=5xyz\left(x^2+y^2+z^2\right)\left(đpcm\right)\)