Question

1. $sqrt[3]{3-sqrt{10}} cdot sqrt[6]{19+6sqrt{10}}$

2. $frac{1}{y} - frac{1}{x}$, если $frac{sqrt{2}y - sqrt{2}x}{xy}=sqrt{32}$

3. $frac{4sqrt[4]{x} + xsqrt{2}}{2sqrt[4]{x} + sqrt{2x}} + sqrt{4+x-4sqrt{x}}$ . В ответ напишите при х=$frac{81}{64}$

Answer 1

$1)sqrt[3]{3-sqrt{10}}*sqrt[6]{19+6sqrt{10}}=sqrt[3]{3-sqrt{10}}*sqrt[6]{9+2*3*sqrt{10}+10}=sqrt[3]{3-sqrt{10}}*sqrt[6]{(3+sqrt{10})^{2}}=sqrt[3]{3-sqrt{10}}*sqrt[3]{3+sqrt{10}} =sqrt[3]{(3-sqrt{10})(3+sqrt{10})}=sqrt[3]{3^{2}-(sqrt{10})^{2}}=sqrt[3]{9-10}=sqrt[3]{-1}=-1$

$2)frac{sqrt{2}y-sqrt{2}x}{xy}=sqrt{32}\\frac{sqrt{2}(y-x) }{xy}=sqrt{32}\\frac{y-x}{xy}=frac{sqrt{32}}{sqrt{2}}\\frac{y-x}{xy}=4\\\frac{1}{y}-frac{1}{x}=frac{x-y}{xy}=-frac{y-x}{xy}=-4$

$3)frac{4sqrt[4]{x}+xsqrt{2}}{2sqrt[4]{x}+sqrt{2x}} +sqrt{4+x-4sqrt{x}}=frac{sqrt{2}*sqrt[4]{x}*(2sqrt{2}+sqrt[4]{x^{3}})}{sqrt{2}*sqrt[4]{x}*(sqrt{2}+sqrt[4]{x})}+sqrt{4-4sqrt{x} +x}=frac{(sqrt{2})^{3}+(sqrt[4]{x})^{3}}{sqrt{2}+sqrt[4]{x}}+sqrt{(2-sqrt{x})^{2}}=frac{(sqrt{2}+sqrt[4]{x})(2-sqrt{2}*sqrt[4]{x}+sqrt[4]{x^{2}})}{sqrt{2}+sqrt[4]{x}}+2-sqrt{x}=2-4sqrt[4]{x}+sqrt{x}+2-sqrt{x}=4-sqrt[4]{4x}$

$x=frac{81}{64} Rightarrow 4-sqrt[4]{4*frac{81}{64}}=4-sqrt[4]{frac{81}{16}}=4-sqrt[4]{(frac{3}{2})^{4}}=4-1,5=2,5$