Решение №16917: \(\left ( \sqrt[3]{\left ( \frac{1}{2} \right )^{-3}}-t^{3}+\sqrt[3]{\frac{t^{5}+2t^{4}+4t^{3}}{4-4t+t^{2}}} \right ):\left ( \frac{1}{\sqrt{2}-\sqrt{t}}+\frac{1}{\sqrt{2}+\sqrt{t}} \right )=\left ( \sqrt[3]{2^{3}-t^{3}}+\sqrt[3]{\frac{t^{3}\left ( t^{2}+2t+4 \right )}{\left ( 2-t \right )^{2}}} \right ):\left ( \frac{\sqrt{t}+\sqrt{2}+\sqrt{2}-\sqrt{t}}{\left ( \sqrt{2}-\sqrt{t} \right )\left ( \sqrt{2}+\sqrt{t} \right )} \right )=\frac{\sqrt[3]{\left ( 2-t \right )^{3}\left ( 4+2t+t^{2} \right )}+t\sqrt[3]{4+2t+t^{2}}}{\sqrt[3]{\left ( 2-t \right )^{2}}}\cdot \frac{2-t}{2\sqrt{2}}=\frac{\sqrt[3]{\left ( 2-t \right )\left ( 4+2t+t^{2} \right )}}{\sqrt{2}}=\frac{\sqrt[3]{8-t^{3}}}{\sqrt{2}}\)

Ответ: \(\frac{\sqrt[3]{8-t^{3}}}{\sqrt{2}}\)

Решение №16918: \(\left ( \left ( \frac{\sqrt{a}-1}{\sqrt{a}+1} \right )^{-1} \left (\frac{\sqrt{a}-1}{\sqrt{a}+1} \right )^{\frac{1}{2}} -\sqrt{a-1}\left ( \sqrt{a}+1 \right )^{-1}\right )\cdot \frac{1}{a^{\frac{2}{3}}+a^{\frac{1}{3}}+1}=\left ( \frac{\sqrt{a}+1}{\sqrt{a}-1}\sqrt{\frac{\sqrt{a}-1}{\sqrt{a}+1}}-\frac{\sqrt{a-1}}{\sqrt{a}+1} \right )^{-2}\cdot \frac{1}{a^{\frac{2}{3}}+a^{\frac{1}{3}}+1}=\left ( \frac{\sqrt{a}+1}{\sqrt{\sqrt{a}-1}}-\frac{\sqrt{\sqrt{a}-1}}{\sqrt{\sqrt{a}+1}} \right )^{-2}\cdot \frac{1}{\sqrt[3]{a^{2}}+\sqrt[3]{a}+1}=\left ( \frac{\sqrt{a}+1-\sqrt{a}+1}{\sqrt{a-1}} \right )^{-2}\cdot \frac{1}{\sqrt[3]{a^{2}}+\sqrt[3]{a}+1}=\left ( \frac{2}{\sqrt{a-1}} \right )^{-2}\cdot \frac{1}{\sqrt[3]{a^{2}}+\sqrt[3]{a}+1}=\frac{a-1}{4\left ( a^{2}+\sqrt[3]{a}+1 \right )}=\frac{\left ( \sqrt[3]{a}-1 \right )\left ( \sqrt[3]{a^{2}}+\sqrt[3]{a}+1 \right )}{4\left ( a^{2}+\sqrt[3]{a}+1 \right )}=\frac{\sqrt[3]{a}-1}{4}\)

Ответ: \(\frac{\sqrt[3]{a}-1}{4}\)

Решение №16919: \(\frac{\sqrt[3]{x^{9}-x^{6}y^{3}}-y^{2}\sqrt[3]{\frac{8x^{6}}{y^{3}}-8x^{3}}+xy\sqrt[3]{y^{3}-\frac{y^{6}}{x^{3}}}}{\sqrt[3]{x^{8}}\left ( x^{2}-2y^{2} \right )+\sqrt[3]{x^{2}y^{12}}}:\frac{\sqrt[3]{1+\frac{y}{x}+\left ( \frac{y}{x} \right )^{2}}}{x+y}=\frac{\sqrt[3]{x^{3}-y^{3}}\left ( x-y \right )^{2}}{\left ( x^{2}-y^{2} \right )}\cdot \frac{x+y}{\sqrt[3]{x^{2}+xy+y^{2}}}=\frac{\sqrt[3]{x-y}\left ( x-y \right )}{x^{2}-y^{2}}=\frac{\sqrt[3]{x-y}}{x+y}\)

Ответ: \(\frac{\sqrt[3]{x-y}}{x+y}\)

Решение №16920: \(\left ( \frac{\left ( 1+a^{-\frac{1}{2}} \right )^{\frac{1}{6}}}{\left ( a^{\frac{1}{2}}+1 \right )^{-\frac{1}{3}}}-\frac{\left ( a^{\frac{1}{2}}-1 \right )^{\frac{1}{3}}}{\left ( 1-a^{-\frac{1}{2}} \right )^{-\frac{1}{6}}} \right )^{-2}\cdot \frac{\frac{1}{3}a^{\frac{1}{12}}}{\sqrt{a}+\sqrt{a-1}}=\left ( \sqrt[6]{\frac{\sqrt{a}+1}{\sqrt{a}}}\sqrt[6]{\left ( \sqrt{a}+1 \right )^{2}}-\sqrt[6]{\left ( \sqrt{a}-1 \right )^{2}}\sqrt[6]{\frac{\sqrt{a}-1}{\sqrt{a}}} \right )^{-2}\frac{\sqrt[12]{a}}{3\left ( \sqrt{a}+\sqrt{a-1} \right )}=\left ( \frac{\sqrt{\sqrt{a}+1}-\sqrt{\sqrt{a}-1}}{\sqrt[12]{a}} \right )^{-2}\cdot \frac{\sqrt[12]{a}}{3\left ( \sqrt{a}+\sqrt{a-1} \right )}=\frac{\sqrt[6]{a}}{2\left ( \sqrt{a}-\sqrt{a-1} \right )}\cdot \frac{\sqrt[12]{a}}{3\left ( \sqrt{a}+\sqrt{a-1} \right )}=\frac{\sqrt[4]{a}}{6\left ( a-a+1 \right )}=\frac{\sqrt[4]{a}}{6}\)

Ответ: \(\frac{\sqrt[4]{a}}{6}\)

Решение №16921: \(\frac{x-y}{x^{\frac{3}{4}}+x^{\frac{1}{2}}y^{\frac{1}{4}}}\cdot \frac{x^{\frac{1}{2}}y^{\frac{1}{4}}+x^{\frac{1}{4}}y^{\frac{1}{2}}}{x^{\frac{1}{2}}+y^{\frac{1}{2}}}\cdot \frac{x^{\frac{1}{4}}y^{-\frac{1}{4}}}{x^{\frac{1}{1}}-2x^{\frac{1}{4}}y^{\frac{1}{4}}+y^{\frac{1}{2}}}=\frac{x-y}{\sqrt[4]{x^{2}}\left ( \sqrt[4]{x}+\sqrt[4]{y} \right )}\cdot \frac{\sqrt[4]{x}\sqrt[4]{y}\left ( \sqrt[4]{x}+\sqrt[4]{y} \right )}{\sqrt{x}+\sqrt{y}}\cdot \frac{\sqrt[4]{x}}{\sqrt[4]{y}\left ( \sqrt[4]{x}-\sqrt[4]{y} \right )^{2}}=\frac{x-y}{\left ( \sqrt{x}+\sqrt{y} \right )\left ( \sqrt[4]{x}-\sqrt[4]{y} \right )^{2}}=\frac{\sqrt[4]{x}+\sqrt[4]{y}}{\sqrt[4]{x}-\sqrt[4]{y}}\)

Ответ: \(\frac{\sqrt[4]{x}+\sqrt[4]{y}}{\sqrt[4]{x}-\sqrt[4]{y}}\)

Решение №16922: \(\left ( \frac{1}{a+\sqrt{2}}-\frac{a^{2}+4}{a^{3}+2\sqrt{2}} \right ):\left ( \frac{a}{2}-\frac{1}{\sqrt{2}}+\frac{1}{a} \right )^{-1}=\left ( \frac{1}{a+\sqrt{2}}-\frac{a^{2}+4}{\left ( a+\sqrt{2} \right )\left ( a^{2}+\sqrt{2}a+2 \right )} \right )\cdot \frac{a^{2}+\sqrt{2}a+2}{2a}=\frac{a^{2}-\sqrt{2}a+2-a^{2}-4}{\left ( a+\sqrt{2} \right )\left ( a^{2}+\sqrt{2}a+2 \right )}\cdot \frac{a^{2}-\sqrt{2}a+2}{2a}=\frac{-\sqrt{2}a-2}{a+\sqrt{2}}\cdot \frac{1}{2a}=\frac{-\sqrt{2}\left ( a+\sqrt{2} \right )}{a+\sqrt{2}}\cdot \frac{1}{\sqrt{2a}}=-\frac{\sqrt{2}}{2a}\)

Ответ: \(-\frac{\sqrt{2}}{2a}\)

Решение №16923: \(\frac{1+z}{1+\sqrt{1+z}}-\frac{1-z}{1-\sqrt{1-z}}=\frac{1-\sqrt{1+z}+z-z\sqrt{1+z}}{1-1-z}-\frac{1+\sqrt{1+z}+z-z\sqrt{1+z}}{1-1+z}=\frac{1+z-\sqrt{1+z}\left ( 1+z \right )}{-z}-\frac{1-z+\sqrt{1-z}\left ( 1-z \right )}{z}=\frac{\sqrt{\left ( 1+z \right )^{3}}-\sqrt{\left ( 1-z \right )^{3}}-2}{z}=\frac{\sqrt{2-2\sqrt{1-z^{2}}}\left ( 2+\sqrt{1-z^{2}} \right )}{z}-\frac{2}{z}=\frac{\sqrt{2-2\sqrt{1-\frac{3}{4}}}\left ( 2+\sqrt{\frac{1}{4}} \right )-2}{\frac{\sqrt{3}}{2}}=\frac{1}{2}:\frac{\sqrt{3}}{2}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}\)

Ответ: \(\frac{\sqrt{3}}{3}\)

Решение №16924: \(\left ( \frac{2-n}{n-1}+4\frac{m-1}{m-2} \right ):\left ( n^{2}\frac{m-1}{n-1}+m^{2}\frac{2-n}{m-2} \right )=\frac{3mn-2\left ( m+n \right )}{\left ( n-1 \right )\left ( m-2 \right )}:\frac{\left ( m-n \right )\left ( 3mn-2\left ( m+n \right ) \right )}{\left ( n-1 \right )\left ( m-2 \right )}=\frac{3mn-2\left ( m+n \right )}{\left ( n-1 \right )\left ( m-2 \right )}\cdot \frac{\left ( n-1 \right )\left ( m-2 \right )}{\left ( m-n \right )\left ( 3mn-2\left ( m+n \right ) \right )}=\frac{1}{m-n}=\frac{1}{\sqrt[4]{400}-\sqrt{5}}=\frac{1}{\sqrt{20}-\sqrt{5}}=\frac{1}{2\sqrt{5}-\sqrt{5}}=\frac{1}{\sqrt{5}}=\frac{\sqrt{5}}{5}\)

Ответ: \(\frac{\sqrt{5}}{5}\)

Решение №16925: \(\frac{\frac{1}{\sqrt{3+x}\sqrt{x+2}}+\frac{1}{\sqrt{3-x}\sqrt{x-2}}}{\frac{1}{\sqrt{3+x}\sqrt{x+2}}-\frac{1}{\sqrt{3-x}\sqrt{x-2}}}; x=\sqrt{6};=\frac{\frac{1}{\sqrt{3+\sqrt{6}}\sqrt{\sqrt{6}+2}}+\frac{1}{\sqrt{3-\sqrt{6}}\sqrt{\sqrt{6}-2}}}{\frac{1}{\sqrt{3+\sqrt{6}}\sqrt{\sqrt{6}+2}}-\frac{1}{\sqrt{3-\sqrt{6}}\sqrt{\sqrt{6}-2}}}=\frac{\sqrt{\left ( 3-\sqrt{6} \right )\left ( \sqrt{6}-2 \right )}+\sqrt{\left ( 3+\sqrt{6} \right )\left ( \sqrt{6}+2 \right )}}{\sqrt{\left ( 3-\sqrt{6} \right )\left ( \sqrt{6}-2 \right )}-\sqrt{\left ( 3+\sqrt{6} \right )\left ( \sqrt{6}+2 \right )}}=\frac{\sqrt{5\sqrt{6}-12}+\sqrt{5\sqrt{6}+12}}{\sqrt{5\sqrt{6}-12}-\sqrt{5\sqrt{6}+12}}=\frac{10\sqrt{6}+2\sqrt{\left ( 5\sqrt{6} \right )^{2}-12^{2}}}{-24}=\frac{5\sqrt{6}+\sqrt{150-144}}{-12}=\frac{5\sqrt{6}+\sqrt{6}}{-12}=-\frac{\sqrt{6}}{2}\)

Ответ: \(-\frac{\sqrt{6}}{2}\)

Решение №16926: \(\left ( \frac{2\left ( a+1 \right )+2\sqrt{a^{2}+2a}}{3a+1-2\sqrt{a^{2}+2a}} \right )^{\frac{1}{2}}-\left ( \sqrt{2a+1}-\sqrt{a} \right )^{-1}\cdot \sqrt{a+2}=\sqrt{\frac{2a+2+2\sqrt{a^{2}+2a}}{3a+1-2\sqrt{2a^{2}+a}}}-\frac{\sqrt{a+2}}{\sqrt{2a+1}-\sqrt{a}}=\sqrt{\frac{\left ( \sqrt{a+2}+\sqrt{a} \right )^{2}}{\left ( \sqrt{2a+1}-\sqrt{a} \right )^{2}}}-\frac{\sqrt{a+2}}{\sqrt{2a+1}-\sqrt{a}}=\frac{\sqrt{a+2}+\sqrt{a}-\sqrt{a+2}}{\sqrt{2a+1}-\sqrt{a}}=\frac{\sqrt{a}}{\sqrt{2a+1}-\sqrt{a}}\)

Ответ: \(\frac{\sqrt{a}}{\sqrt{2a+1}-\sqrt{a}}\)