Question

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Answer 1

$\frac{x-1}{x+x^{ \frac{1}{2}}+1 } : \frac{x^{0,5}+1}{x^{1,5}-1} + \frac{2}{x^{-0,5}} = \frac{x^{2,5}-x-x^{2,5}+1}{(x+x^{ \frac{1}{2} }+1)(x^{ \frac{1}{2} }+1)} +2x^{ \frac{1}{2} }= \frac{(\sqrt{x^5}-x-\sqrt{x^3}+1}{(x+\sqrt{x}+1)(\sqrt{x}+1)} +\\+2\sqrt{x}= \frac{x(x\sqrt{x}-1)-(x\sqrt{x}-1)}{(x+\sqrt{x}+1)(\sqrt{x}+1)} +2\sqrt{x}= \frac{x(\sqrt{x}-1)(x+\sqrt{x}+1)-(\sqrt{x}-1)(x+\sqrt{x}+1)}{(x+\sqrt{x}+1)(\sqrt{x}+1)\+2\sqrt{x}}\\+2\sqrt{x}= \frac{(\sqrt{x}-1)(\sqrt{x}-1)(\sqrt{x}+1)}{\sqrt{x}+1}$
$+2\sqrt{x}=(\sqrt{x}-1)(\sqrt{x}-)+2\sqrt{x}=(\sqrt{x}-1)^2+2\sqrt{x}=x-2\sqrt{x}+1+\\+2\sqrt{x}=x+1$

Answer 2

$\mathtt{\frac{x-1}{x+x^{\frac{1}{2}}+1}*\frac{x^{\frac{3}{2}}-1}{x^{\frac{1}{2}}+1}+2x^{\frac{1}{2}}=\frac{(x-1)(x^{\frac{3}{2}}-1)}{(x+x^{\frac{1}{2}}+1)(x^{\frac{1}{2}}+1)}+2x^{\frac{1}{2}}=\frac{(x-1)(x^{\frac{1}{2}}-1)}{x^{\frac{1}{2}}+1}+2x^{\frac{1}{2}}=}\\\mathtt{(x^{\frac{1}{2}}-1)^2+2x^{\frac{1}{2}}=x-2x^{\frac{1}{2}}+1+2x^{\frac{1}{2}}=x+1}$