Mam problem z takimi przykładami :
\(1. \frac{1}{(x+y)^2} ( \frac{1}{x^2}+ \frac{1}{y^2})+ \frac{2}{(x+y)^3}( \frac{1}{x}+ \frac{1}{y})
2. \frac{x^2+y^2}{x^2-xy}( \frac{x}{y}-1)* \frac{(x+y)-(x-y)}{(x+y)^2+(x-y)^2}
3. ( \frac{a-b}{a+b}+ \frac{a+b}{a-b}) + (\frac{a^2+b^2}{2ab}+1)* \frac{ab}{a^2+b^2}
4. \frac{(a+x)^{-1}+(a-x)^{-1}}{(a-x)^{-1}-(a+x)^{-1}}* \frac{3( \sqrt{ \frac{x}{a} }- \sqrt{ \frac{a}{x} } ) }{2 (\sqrt{ \frac{x}{a} }+ \sqrt{ \frac{a}{x} } )}\)
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1.
\(\frac{1}{(x+y)^2}\cdot(\frac{1}{x^2}+\frac{1}{y^2})+\frac{2}{(x+y)^3}\cdot(\frac{1}{x}+\frac{1}{y})=\frac{1}{(x+y)^2}\cdot\frac{x^2+y^2}{x^2y^2}+\frac{2}{(x+y)^3}\cdot\frac{x+y}{xy}=\\=\frac{x^2+y^2}{x^2y^2(x+y)^2}+\frac{2(x+y)}{xy(x+y)^3}=\\=\frac{x^2+y^2}{x^2y^2(x+y)^2}+\frac{2}{xy(x+y)^2}=\frac{x^2+y^2}{x^2y^2(x+y)^2}+\frac{2xy}{x^2y^2(x+y)^2}=\\=\frac{x^2+y^2+2xy}{x^2y^2(x+y)^2}=\frac{(x+y)^2}{x^2y^2(x+y)^2}=\frac{1}{x^2y^2}\)
Oczywiście, trzeba napisać zastrzeżenia:
\(x\neq0\ \wedge\ y\neq0\ \wedge\ y\neq-x\)
\(\frac{1}{(x+y)^2}\cdot(\frac{1}{x^2}+\frac{1}{y^2})+\frac{2}{(x+y)^3}\cdot(\frac{1}{x}+\frac{1}{y})=\frac{1}{(x+y)^2}\cdot\frac{x^2+y^2}{x^2y^2}+\frac{2}{(x+y)^3}\cdot\frac{x+y}{xy}=\\=\frac{x^2+y^2}{x^2y^2(x+y)^2}+\frac{2(x+y)}{xy(x+y)^3}=\\=\frac{x^2+y^2}{x^2y^2(x+y)^2}+\frac{2}{xy(x+y)^2}=\frac{x^2+y^2}{x^2y^2(x+y)^2}+\frac{2xy}{x^2y^2(x+y)^2}=\\=\frac{x^2+y^2+2xy}{x^2y^2(x+y)^2}=\frac{(x+y)^2}{x^2y^2(x+y)^2}=\frac{1}{x^2y^2}\)
Oczywiście, trzeba napisać zastrzeżenia:
\(x\neq0\ \wedge\ y\neq0\ \wedge\ y\neq-x\)
4.
\(\frac{(a+x)^{-1}+(a-x)^{-1}}{(a-x)^{-1}-(a+x)^{-1}}\cdot\frac{3(\sqrt{\frac{x}{a}}-\sqrt{\frac{a}{x}})}{2(\sqrt{\frac{x}{a}}+\sqrt{\frac{a}{x}})}=\frac{\frac{1}{a+x}+\frac{1}{a-x}}{\frac{1}{a-x}-\frac{1}{a+x}}{\frac{1}{a-x}-\frac{1}{a+x}}\cdot\frac{3(\frac{\sqrt{x}}{\sqrt{a}}-\frac{\sqrt{a}}{\sqrt{x}})}{2(\frac{\sqrt{x}}{\sqrt{a}}+\frac{\sqrt{a}}{\sqrt{x}})}=\\=\frac{\frac{a-x+a+x}{(a+x)(a-x)}}{\frac{a+x-a+x}{(a-x)(a+x)}}\cdot\frac{3\cdot\frac{x-a}{\sqrt{ax}}}{2\cdot\frac{x+a}{\sqrt{ax}}}=\frac{2a}{2x}\cdot\frac{3(x-a)}{2(x+a)}=\frac{3a(x-a)}{2x(x+a)}\)
\(\frac{(a+x)^{-1}+(a-x)^{-1}}{(a-x)^{-1}-(a+x)^{-1}}\cdot\frac{3(\sqrt{\frac{x}{a}}-\sqrt{\frac{a}{x}})}{2(\sqrt{\frac{x}{a}}+\sqrt{\frac{a}{x}})}=\frac{\frac{1}{a+x}+\frac{1}{a-x}}{\frac{1}{a-x}-\frac{1}{a+x}}{\frac{1}{a-x}-\frac{1}{a+x}}\cdot\frac{3(\frac{\sqrt{x}}{\sqrt{a}}-\frac{\sqrt{a}}{\sqrt{x}})}{2(\frac{\sqrt{x}}{\sqrt{a}}+\frac{\sqrt{a}}{\sqrt{x}})}=\\=\frac{\frac{a-x+a+x}{(a+x)(a-x)}}{\frac{a+x-a+x}{(a-x)(a+x)}}\cdot\frac{3\cdot\frac{x-a}{\sqrt{ax}}}{2\cdot\frac{x+a}{\sqrt{ax}}}=\frac{2a}{2x}\cdot\frac{3(x-a)}{2(x+a)}=\frac{3a(x-a)}{2x(x+a)}\)
3.
\((\frac{a-b}{a+b}+\frac{a+b}{a-b})\cdot(\frac{a^2+b^2}{2ab}+1)\cdot\frac{ab}{a^2+b^2}=\frac{(a-b)^2+(a+b)^2}{(a+b)(a-b)}\cdot\frac{a^2+b^2+2ab}{2ab}\cdot\frac{ab}{a^2+b^2}=\\=\frac{a^2-2ab+b^2+a^2+2ab+b^2}{(a+b)(a-b)}\cdot\frac{(a+b)^2}{2(a^2+b^2)}=\\=\frac{2(a^2+b^2)}{(a+b)(a-b)}\cdot\frac{(a+b)^2}{2(a^2+b^2)}=\frac{a+b}{a-b}\)
\((\frac{a-b}{a+b}+\frac{a+b}{a-b})\cdot(\frac{a^2+b^2}{2ab}+1)\cdot\frac{ab}{a^2+b^2}=\frac{(a-b)^2+(a+b)^2}{(a+b)(a-b)}\cdot\frac{a^2+b^2+2ab}{2ab}\cdot\frac{ab}{a^2+b^2}=\\=\frac{a^2-2ab+b^2+a^2+2ab+b^2}{(a+b)(a-b)}\cdot\frac{(a+b)^2}{2(a^2+b^2)}=\\=\frac{2(a^2+b^2)}{(a+b)(a-b)}\cdot\frac{(a+b)^2}{2(a^2+b^2)}=\frac{a+b}{a-b}\)