[tex]\displaystyle\bf\\\frac{\sqrt{x} +1}{x\sqrt{x} +x+\sqrt{x} } :\frac{x-1}{x^{2} -\sqrt{x} }=\\\\\\=\frac{\sqrt{x} +1}{(x\cdot \sqrt{x} +(\sqrt{x} )^{2} +\sqrt{x} } :\frac{(\sqrt{x} )^{2} -1^{2} }{[(\sqrt{x} )^{2} ]^{2} -\sqrt{x} }=\\\\\\=\frac{\sqrt{x} +1}{\sqrt{x} \cdot(x+\sqrt{x} +1)} :\frac{(\sqrt{x} -1)\cdot(\sqrt{x} +1)}{\sqrt{x} \cdot[(\sqrt{x} )^{3}-1] } =[/tex]
[tex]\displaystyle\bf\\=\frac{\sqrt{x} +1}{\sqrt{x} \cdot(x+\sqrt{x} +1)}\cdot\frac{\sqrt{x} \cdot(\sqrt{x} -1)\cdot(x+\sqrt{x} +1)}{(\sqrt{x} -1)\cdot(\sqrt{x} +1)} =1[/tex]
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[tex]\displaystyle\bf\\\frac{\sqrt{x} +1}{x\sqrt{x} +x+\sqrt{x} } :\frac{x-1}{x^{2} -\sqrt{x} }=\\\\\\=\frac{\sqrt{x} +1}{(x\cdot \sqrt{x} +(\sqrt{x} )^{2} +\sqrt{x} } :\frac{(\sqrt{x} )^{2} -1^{2} }{[(\sqrt{x} )^{2} ]^{2} -\sqrt{x} }=\\\\\\=\frac{\sqrt{x} +1}{\sqrt{x} \cdot(x+\sqrt{x} +1)} :\frac{(\sqrt{x} -1)\cdot(\sqrt{x} +1)}{\sqrt{x} \cdot[(\sqrt{x} )^{3}-1] } =[/tex]
[tex]\displaystyle\bf\\=\frac{\sqrt{x} +1}{\sqrt{x} \cdot(x+\sqrt{x} +1)}\cdot\frac{\sqrt{x} \cdot(\sqrt{x} -1)\cdot(x+\sqrt{x} +1)}{(\sqrt{x} -1)\cdot(\sqrt{x} +1)} =1[/tex]