Решение №17067: \(\left ( \frac{1}{\sqrt{a}+\sqrt{a+1}} +\frac{1}{\sqrt{a}-\sqrt{a-1}}\right ):\left ( 1+\sqrt{\frac{a+1}{a-1}} \right )=\frac{\sqrt{a}-\sqrt{a-1}+\sqrt{a}+\sqrt{a+1}}{\left ( \sqrt{a}+\sqrt{a+1} \right )\left (\sqrt{a}-\sqrt{a-1} \right )}:\frac{\sqrt{a+1}+\sqrt{a-1}}{\sqrt{a-1}}=\frac{2\sqrt{a}+\sqrt{a+1}-\sqrt{a-1}}{a+\sqrt{a\left ( a+1 \right )}-\sqrt{a\left ( a-1 \right )}-\sqrt{\left ( a-1 \right )\left ( a+1 \right )}}:\frac{\sqrt{a+1}+\sqrt{a-1}}{\sqrt{a-1}}=\frac{\sqrt{a-1}\left ( 2\sqrt{a}+\sqrt{a+1}-\sqrt{a-1} \right )}{2\sqrt{a}+\sqrt{a+1}-\sqrt{a-1}}=\sqrt{a-1}\)

Ответ: \(\sqrt{a-1}\)

Решение №17068: \(\frac{\sqrt{3}\left ( a-b^{2} \right )+\sqrt{3}b\sqrt[3]{8b^{3}}}{\sqrt{2\left ( a-b^{2} \right )^{2}+\left ( 2b\sqrt{2a} \right )^{2}}}\cdot \frac{\sqrt{2a}-\sqrt{2c}}{\sqrt{\frac{3}{a}-\sqrt{\frac{3}{c}}}}=\frac{\sqrt{3}\left ( a-b^{2}+2b^{2} \right )}{\sqrt{2}\sqrt{a^{2}-2ab^{2}+b^{4}+4ab^{2}}}\cdot \frac{\sqrt{2}\left ( \sqrt{a}-\sqrt{c} \right )\sqrt{ac}}{\sqrt{3}\left ( \sqrt{c}-\sqrt{a} \right )}=\frac{a+b^{2}}{\sqrt{a^{2}+2ab^{2}+b}}\cdot \frac{-\sqrt{ac}}{1}=\frac{-\left ( a+b^{2} \right )\sqrt{ac}}{\sqrt{\left ( a+b^{2} \right )^{2}}}=-\sqrt{ac}\)

Ответ: \(-\sqrt{ac}\)

Решение №17069: \(\frac{\left ( x\sqrt[4]{x}-\sqrt{xy}\left ( \sqrt[4]{x}-\sqrt[4]{y} \right )-y\sqrt[4]{y} \right )\left ( x+y+\sqrt{xy} \right )}{\left ( \sqrt[4]{x}+\sqrt[4]{y} \right )\left ( \left ( \sqrt[4]{x}-\sqrt[4]{y} \right )^{2} +\sqrt[4]{xy}\right )}=\frac{\left ( \sqrt[4]{x^{2}}-\sqrt[4]{y^{2}} \right )\left ( \sqrt[4]{x^{3}}+\sqrt[4]{y^{3}} \right )\left ( \sqrt[4]{x^{4}}+\sqrt[4]{x^{2}y^{2}}+\sqrt[4]{y^{4}} \right )}{\sqrt[4]{x^{3}}+\sqrt[4]{y^{3}}}=\left ( \sqrt[4]{x^{2}}-\sqrt[4]{y^{2}} \right )\left ( \sqrt[4]{x^{4}}+\sqrt[4]{x^{2}y^{2}}+\sqrt[4]{y^{4}} \right )=\left ( \sqrt{x}-\sqrt{y} \right )\left ( \sqrt{x^{2}}+\sqrt{xy}+\sqrt{y^{2}} \right )=\sqrt{x^{3}}-\sqrt{y^{3}}\)

Ответ: \(\sqrt{x^{3}}-\sqrt{y^{3}}\)

Решение №17070: \(\frac{\sqrt{\left ( x+2 \right )^{2}-8x}}{\sqrt{x}-\frac{2}{\sqrt{x}}}=\frac{\sqrt{x^{2}+4x+4-8x}}{\frac{\sqrt{x}^{2}-2}{\sqrt{x}}}=\frac{\sqrt{x}\sqrt{x^{2}-4x-4}}{x-2}=\frac{\sqrt{x}\sqrt{\left ( x^{2}-2 \right )^{2}}}{x-2}=\frac{\sqrt{x}\left | x-2 \right |}{x-2}\)

Ответ: \(\sqrt{x},-\sqrt{x}\)

Решение №17071: \(\frac{\sqrt{\left ( 3x+2 \right )^{2}-24x}}{3\sqrt{x}-\frac{2}{\sqrt{x}}}=\frac{\sqrt{9x^{2}+12x+4-24x}}{\frac{3x-2}{\sqrt{x}}}=\frac{\sqrt{9x^{2}-12x+4}\sqrt{x}}{3x-2}=\frac{\sqrt{\left ( 3x-2 \right )^{2}}\sqrt{x}}{3x-2}=\frac{\left | 3x-2 \right |\sqrt{x}}{3x-2}=-\sqrt{x};\sqrt{x}\)

Ответ: \(-\sqrt{x};\sqrt{x}\)

Решение №17072: \(\frac{x-1}{x^{\frac{3}{4}}+x^{\frac{1}{2}}}\cdot \frac{x^{\frac{1}{2}}+x\frac{1}{4}}{x^{\frac{1}{2}}+1}\cdot x^{\frac{1}{4}}+1=\frac{x^{\frac{1}{2}}-1}{x^{\frac{2}{4}}\left ( x^{\frac{1}{4}}+1 \right )}\cdot \frac{x^{\frac{1}{4}}\left ( x^{\frac{1}{4}}+1 \right )}{1}\cdot x^{\frac{1}{4}}+1=x^{\frac{1}{2}}-1+1=x^{\frac{1}{2}}=\sqrt{x}\)

Ответ: \(\sqrt{x}\)

Решение №17073: \(\frac{1-x^{-2}}{x^{\frac{1}{2}}-x^{-\frac{1}{2}}}-\frac{2}{x^{\frac{3}{2}}}+\frac{x^{-2}-x}{x^{\frac{1}{2}}-x^{-\frac{1}{2}}}=\frac{1-\frac{1}{x^{2}}}{\sqrt{x}-\frac{1}{\sqrt{x}}}+\frac{\frac{1}{x^{2}}-x}{\sqrt{x}-\frac{1}{\sqrt{x}}}-\frac{2}{\sqrt{x^{3}}}=\frac{\left ( 1-x \right )\sqrt{x}}{x-1}-\frac{2}{\sqrt{x^{3}}}=-\sqrt{x}-\frac{2}{\sqrt{x^{3}}}=\frac{-\sqrt{x}\sqrt{x^{3}}-2}{\sqrt{x^{3}}}=\frac{-\sqrt{x^{4}}-2}{\sqrt{x^{3}}}=\frac{-x^{2}-2}{\sqrt{x^{3}}}=-\frac{x^{2}+2}{x\sqrt{x}}=-\frac{\left ( x^{2}+2 \right )\sqrt{x}}{x\sqrt{x\sqrt{x}}}=-\sqrt{x}\left ( 1+\frac{2}{x^{2}} \right )\)

Ответ: \(-\sqrt{x}\left ( 1+\frac{2}{x^{2}} \right )\)

Решение №17074: \(x=\sqrt[3]{4+\sqrt{80}}-\sqrt[3]{\sqrt{80}-4}=\sqrt[3]{4+\sqrt{80}}+\sqrt[3]{4-\sqrt{80}};\left ( \sqrt[3]{4+\sqrt{80}}+\sqrt[3]{4-\sqrt{80}} \right )^{3}+12\left ( \sqrt[3]{4+\sqrt{80}}+\sqrt[3]{4-\sqrt{80}} \right )-8=0;\left ( \sqrt[3]{4+\sqrt{80}} \right )^{3}+3\left ( \sqrt[3]{4+\sqrt{80}} \right )^{2}\cdot \sqrt[3]{4-\sqrt{80}}+3\sqrt[3]{4+\sqrt{80}}\cdot \left ( \sqrt[3]{4-\sqrt{80}} \right )^{2}+\left ( \sqrt[3]{4+\sqrt{80}} \right )^{3}+12\left ( \sqrt[3]{4+\sqrt{80}}+\sqrt[3]{4-\sqrt{80}} \right )-8=0;4+\sqrt{80}+3\sqrt[3]{\left ( 4+\sqrt{80} \right )\left ( 4-\sqrt{80} \right )}\left ( \sqrt[3]{4+\sqrt{80}}+\sqrt[3]{4-\sqrt{80}} \right )+4-\sqrt{80}+12\left ( \sqrt[3]{4+\sqrt{80}}+\sqrt[3]{4-\sqrt{80}} \right )-8=0;4+\sqrt{80}-12\left ( \sqrt[3]{4+\sqrt{80}}+\sqrt[3]{4-\sqrt{80}} \right )+4-\sqrt{80}+12\left ( \sqrt[3]{4+\sqrt{80}}+\sqrt[3]{4-\sqrt{80}} \right )-8=0;0=0\)

Ответ: \(0=0\)

Решение №17075: \(\frac{\sqrt{x-2\sqrt{x-1}}}{\sqrt{x-1}-1}=\frac{\sqrt{x-1-2\sqrt{x-1}+1}}{\sqrt{x-1}-1}=\frac{\sqrt{\left ( \sqrt{x-1} \right )^{2}-2\sqrt{x-1}+1}}{\sqrt{x-1}-1}=\frac{\sqrt{\left ( \sqrt{x-1} -1\right )^{2}}}{\sqrt{x-1}-1}=\frac{\left | \sqrt{x-1}-1 \right |}{\sqrt{x-1}-1}=-1;1\)

Ответ: \(-1;1\)

Решение №17076: \(\left ( 4x-1 \right )\left ( \frac{1}{8x}\left ( \sqrt{8x-1}+4x \right )^{-1}-\left ( \sqrt{8x-1}-4x \right )^{-1} \right )^{\frac{1}{2}}=\left ( 4x-1 \right )\sqrt{\frac{\sqrt{8x-1}-4x-\sqrt{8x-1}-4x}{8x\left ( \sqrt{8x-1}+4x \right )\left ( \sqrt{8x-1}-4x \right )}}=\left ( 4x-1 \right )\sqrt{\frac{-1}{8x-1-16x^{2}}}=\left ( 4x-1 \right )\sqrt{\frac{1}{\left ( 4x-1 \right )^{2}}}=\frac{4x-1}{\left | 4x-1 \right |}=-1;1\)

Ответ: \(-1;1\)