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a: \(A=\dfrac{\sqrt{x}+x\sqrt{y}+\sqrt{y}+y\sqrt{x}+\sqrt{x}-x\sqrt{y}-\sqrt{y}+y\sqrt{x}}{1-xy}:\dfrac{1-xy+x+y+2xy}{1-xy}\)

\(=\dfrac{2\sqrt{x}+2y\sqrt{x}}{x+y+xy+1}\)

\(=\dfrac{2\sqrt{x}\left(y+1\right)}{\left(x+1\right)\left(y+1\right)}=\dfrac{2\sqrt{x}}{x+1}\)

b: \(x=\dfrac{1}{\sqrt{2}+1}=\sqrt{2}-1\)

\(A=\dfrac{2\sqrt{\sqrt{2}-1}}{\sqrt{2}-1+1}=\sqrt{2\left(\sqrt{2}-1\right)}\)

a) ( x + y)( x^2 - xy+  y^2 )- ( x - y)( x^2 + xy + y^2 )

= x^3 +y^3 - ( x^3 - y^3 )

= x^3 + y^3 - x^3 + y^3 

= 2y^3 

b; ( x - y)^2 + ( x + y)^2

= x^2 - 2xy + y^2 + x^2 + 2xy + y^2

= 2x^2 + 2y^2 

Hên xui thôi ( cái này không có chắc lắm )

\(\frac{x^3-xy^3+y^3z-yz^3+z^3x-x^3z}{x^2y-xy^2+y^2z-yz^2+z^2x-zx^2}\)

\(=xy-xy+xy-yz+zx-x^3\)\(z\)\(-\)\(zx^2\)

\(=xy-yz-zx-x^3\)\(z\)

phần trên sai rồi cho xin lỗi  ( trình bày lại )

bạn ghi lại đề nha

= xy - xy + yz - yz + zx - x^3z - zx^2

= -zx - x^3z

\(x^2-1=\frac{1}{4}\left(a^2+\frac{1}{a^2}+2\right)-1=\frac{1}{4}\left(a^2+\frac{1}{a^2}-2\right)=\frac{1}{4}\left(a-\frac{1}{a}\right)^2\)

Tương tự \(y^2-1=\frac{1}{4}\left(b-\frac{1}{b}\right)^2\)

\(P=\frac{\frac{1}{4}\left(a+\frac{1}{a}\right)\left(b+\frac{1}{b}\right)-\frac{1}{4}\left(a-\frac{1}{a}\right)\left(b-\frac{1}{b}\right)}{\frac{1}{4}\left(a+\frac{1}{a}\right)\left(b+\frac{1}{b}\right)+\frac{1}{4}\left(a-\frac{1}{a}\right)\left(b-\frac{1}{b}\right)}\)

\(=\frac{ab+\frac{a}{b}+\frac{b}{a}+\frac{1}{ab}-ab+\frac{a}{b}+\frac{b}{a}-\frac{1}{ab}}{ab+\frac{a}{b}+\frac{b}{a}+\frac{1}{ab}+ab-\frac{a}{b}-\frac{b}{a}+\frac{1}{ab}}=\frac{\frac{a}{b}+\frac{b}{a}}{ab+\frac{1}{ab}}=\frac{a^2+b^2}{a^2b^2+1}\)

\(\left(x^2-xy+y^2\right)\left(x^2+xy+y^2\right)\)

\(=\left(x^2\right)^2-\left(xy+y^2\right)^2\)

\(=x^4-\left(x^2y^2+2xy^3+y^4\right)\)

\(=x^4-x^2y^2-2xy^3-y^4\)

P/s: k chắc lắm

ĐKXĐ:...

Để gõ công thức cho nhanh ta đặt \(\left\{{}\begin{matrix}\sqrt{x}=a\\\sqrt{y}=b\end{matrix}\right.\)

\(\frac{a^2+b^2}{a+b}:\left(\frac{a^2+b^2}{ab}+\frac{b^2}{a^2-ab}-\frac{a^2}{b^2+ab}\right)=\frac{a^2+b^2}{ab}:\left(\frac{a^2+b^2}{ab}+\frac{b^2}{a\left(a-b\right)}-\frac{a^2}{b\left(a+b\right)}\right)\)

\(=\frac{a^2+b^2}{ab}:\left(\frac{\left(a^2+b^2\right)\left(a^2-b^2\right)+b^3\left(a+b\right)-a^3\left(a-b\right)}{ab\left(a-b\right)\left(a+b\right)}\right)\)

\(=\frac{a^2+b^2}{ab}:\left(\frac{a^4-b^4+ab^3+b^4-a^4+a^3b}{ab\left(a-b\right)\left(a+b\right)}\right)\)

\(=\frac{a^2+b^2}{ab}:\left(\frac{ab\left(a^2+b^2\right)}{ab\left(a-b\right)\left(a+b\right)}\right)=\frac{\left(a^2+b^2\right)\left(a-b\right)\left(a+b\right)}{a^2+b^2}=a^2-b^2=x-y\)

1)a)=>x²+y²+2xy-4(x²-y²-2xy)

=>x²+y²+2xy-4.x²+4y²+8xy

=>-3.x²+5y²+10xy