Решение №16912: \(\frac{2-\sqrt{2}-\sqrt{3}}{2+\sqrt{2}-\sqrt{3}}=\frac{\sqrt{4}-\sqrt{2}-\sqrt{3}}{\sqrt{4}+\sqrt{2}-\sqrt{3}}=\frac{\left ( \sqrt{4}-\sqrt{2}-\sqrt{3} \right )\left ( \sqrt{4}+\sqrt{2}+\sqrt{3} \right )\left ( 4+2-3-2\sqrt{4*2} \right )}{\left ( \sqrt{4}+\sqrt{2}-\sqrt{3} \right )\left ( \sqrt{4}+\sqrt{2}+\sqrt{3} \right )\left ( 4+2-3-2\sqrt{4*2} \right )}=\frac{\left ( 4-\left ( 2+2\sqrt{6}+3 \right ) \right )\left ( 3-4\sqrt{2} \right )}{9-32}=\frac{\left ( 2\sqrt{6}+1 \right )\left ( 3-4\sqrt{2} \right )}{23}\)

Ответ: \(\frac{\left ( 2\sqrt{6}+1 \right )\left ( 3-4\sqrt{2} \right )}{23}\)

Решение №16913: \(\frac{3+\sqrt{2}+\sqrt{3}}{3-\sqrt{2}-\sqrt{3}}=\frac{\left ( 3+\left ( \sqrt{2}+\sqrt{3} \right ) \right )\left ( 3+\left ( \sqrt{2}+\sqrt{3} \right ) \right )}{\left ( 3-\left ( \sqrt{2}+\sqrt{3} \right ) \right )\left ( 3+\left ( \sqrt{2}+\sqrt{3} \right ) \right )}=\frac{14+6\left ( \sqrt{2}+\sqrt{3} \right )+2\sqrt{6}}{4-2\sqrt{6}}=\frac{20+13\sqrt{3}+15\sqrt{2}+9\sqrt{6}}{-2}=\frac{\left ( 4+3\sqrt{2} \right )\left ( 5+3\sqrt{3} \right )}{2}\)

Ответ: \(\frac{\left ( 4+3\sqrt{2} \right )\left ( 5+3\sqrt{3} \right )}{2}\)

Решение №16914: \(\left ( 2-\frac{1}{4a^{-1}}-\frac{4}{a} \right )\left ( \left ( a-4 \right )\sqrt[3]{\left ( a+4 \right )^{-3}}-\frac{\left ( a+4 \right )^{\frac{3}{2}}}{\sqrt{\left ( a^{2}-16 \right )\left ( a-4 \right )}} \right )=\left ( 2-\frac{a}{4}-\frac{4}{a} \right )\left ( \frac{a-4}{a+4}-\frac{\sqrt{\left ( a+4 \right )^{2}}}{\sqrt{\left ( a+4 \right )\left ( a-4 \right )^{2}}} \right )=\frac{a^{2}-8a+16}{4a}\left ( \frac{a-4}{a+4}-\frac{a+4}{\left | a-4 \right |} \right )=\frac{\left ( a-4 \right )^{2}}{4a}\left ( \frac{a+4}{\left | a-4 \right |}-\frac{a-4}{a+4} \right )=\frac{+}{}\frac{\left ( a-4 \right )^{2}\left ( \left ( a+4 \right )^{2}+\left ( a-4 \right )^{2} \right )}{4a\left ( a-4 \right )\left ( a+4 \right )}=\frac{\left ( 4-a \right )\left ( a^{2}+16 \right )}{2a\left ( a+4 \right )}; \frac{4a-16}{a+4}\)

Ответ: \(\frac{\left ( 4-a \right )\left ( a^{2}+16 \right )}{2a\left ( a+4 \right )}; \frac{4a-16}{a+4}\)

Решение №16915: \(\frac{a^{3}-3a^{2}+4+\left ( a^{2}-4 \right )\sqrt{a^{2}-1}}{a^{3}+3a^{2}-4+\left ( a^{2}-4 \right )\sqrt{a^{2}-1}}=\frac{a^{2}\left ( a-1 \right )-\left ( a^{2}-4 \right )+\left ( a^{2}-4 \right )\sqrt{a^{2}-1}}{a^{2}\left ( a+2 \right )+\left ( a^{2}-4 \right )+\left ( a^{2}-4 \right )\sqrt{a^{2}-1}}=\frac{\left ( a-2 \right )\left ( a-2 \right )^{2}+\left ( a^{2}-4 \right )\sqrt{a^{2}-1}}{\left ( a-1 \right )\left ( a+2 \right )^{2}+\left ( a^{2}-4 \right )\sqrt{a^{2}-1}}=\frac{\left ( a-2 \right )\sqrt{a+1}\left ( \sqrt{a+1}\left ( a-2 \right )+\left ( a+2 \right )\sqrt{a-1} \right )}{\left ( a+2 \right )\sqrt{a-1}\left ( \sqrt{a+1}\left ( a-2 \right )+\left ( a+2 \right )\sqrt{a-1} \right )}=\frac{\left ( a-2 \right )\sqrt{a+1}}{\left ( a+2 \right )\sqrt{a-1}}\)

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

Решение №16916: \(\left ( \left ( z-3 \right )\left ( z+3 \right )^{-1}-\frac{\left ( z+3 \right )^{\frac{3}{2}}}{\sqrt{\left ( z^{2}-9 \right )\left ( z-3 \right )}} \right )\frac{\frac{1}{3}-\frac{z}{18}-\frac{1}{2z}}{\left ( z+3 \right )^{-1}}=\left ( \frac{z-3}{z+3}-\frac{\sqrt{\left ( z+3 \right )^{3}}}{\sqrt{\left ( z-3 \right )^{2}\left ( z+3 \right )}} \right )\cdot \frac{\frac{6z-z^{2}-9}{18z}}{\frac{1}{z+3}}=\left ( \frac{z-3}{z+3}-\frac{z+3}{\left | z-3 \right |} \right )\cdot \frac{-\left ( z-3 \right )^{2}\left ( z+3 \right )}{18z}=\frac{\left ( z-3 \right )^{2}+\left ( z+3 \right )^{2}}{\left ( z+3 \right )\left ( z-3 \right )}\cdot \frac{-\left ( z-3 \right )\left ( z+3 \right )}{18z}=\frac{\left ( z^{2}+9 \right )\left ( 3-z \right )}{9z};\frac{2\left ( z-3 \right )}{3}\)

Ответ: \(\frac{\left ( z^{2}+9 \right )\left ( 3-z \right )}{9z};\frac{2\left ( z-3 \right )}{3}\)

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

Решение №16927: \(\sqrt[4]{\left ( 1-2a+a^{2} \right )\left ( a^{2}-1 \right )\left ( a-1 \right )}:\frac{a^{2}+2a-3}{\sqrt[4]{a+1}}=\sqrt[4]{\left ( a-1 \right )^{2}\left ( a-1 \right )\left ( a+1 \right )\left ( a-1 \right )}\cdot \frac{\sqrt[4]{a+1}}{a^{2}+2a-3}=\frac{\sqrt[4]{\left ( a-1 \right )^{4}\left ( a+1 \right )^{2}}}{\left ( a-1 \right )\left ( a+3 \right )}=\frac{\left | a-1 \right |\sqrt[4]{\left ( a+1 \right )^{2}}}{\left ( a-1 \right )\left ( a+3 \right )}=\frac{\left | a-1 \right |\sqrt{\left ( a+1 \right )}}{\left ( a-1 \right )\left ( a+3 \right )}=-\frac{\sqrt{a+1}}{a+3};\frac{\sqrt{a+1}}{a+3}\)

Ответ: \(-\frac{\sqrt{a+1}}{a+3};\frac{\sqrt{a+1}}{a+3}\)

Решение №16928: \(\frac{\sqrt{2a+2\sqrt{a^{2}-b^{2}}}-\sqrt{a-b}}{\sqrt{2a-2\sqrt{a^{2}-b^{2}}}+\sqrt{a-b}}=\frac{\sqrt{a+b}+\sqrt{a-b}-\sqrt{a-b}}{\left | \sqrt{a+b}-\sqrt{a-b} \right |+\sqrt{a-b}}=\frac{\sqrt{a+b}}{\left | \sqrt{a+b}-\sqrt{a-b} \right |+\sqrt{a-b}}=\frac{\sqrt{a+b}}{2\sqrt{a-b}-\sqrt{a+b}};\frac{\sqrt{a+b}}{\sqrt{a+b}}=\frac{\sqrt{a+b}}{2\sqrt{a-b}-\sqrt{a+b}};1\)

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

Решение №16929: \(\frac{\left ( m-1 \right )\sqrt{m}-\left ( n-1 \right )\sqrt{n}}{\sqrt{m^{3}n}+mn+m^{2}-m}=\frac{m\sqrt{m}-\sqrt{m}-n\sqrt{n}+\sqrt{n}}{m\sqrt{mn}+mn+m^{2}-m}=\frac{\left ( \sqrt{m^{3}}-\sqrt{n^{3}} \right )-\left ( \sqrt{m}-\sqrt{n} \right )}{m\left ( \sqrt{mn}+n+m-1 \right )}=\frac{\left ( \sqrt{m}-\sqrt{n} \right )\left ( m+\sqrt{mn}+n-1 \right )}{m\left ( \sqrt{mn}+n+m-1 \right )}=\frac{\sqrt{m}-\sqrt{n}}{m}\)

Ответ: \(\frac{\sqrt{m}-\sqrt{n}}{m}\)

Решение №16930: \(\frac{\sqrt{\frac{p^{4}+q^{4}}{p^{4}-p^{2}q^{2}}+\frac{2q^{2}}{p^{2}-q^{2}}\left ( p^{3}-pq^{2} \right )-2q\sqrt{p}}}{\sqrt{\frac{p}{p-q}}-\frac{q}{p+q}-\frac{2pq}{p^{2}-q^{2}}\cdot \left ( p-q \right )}=\frac{p-q}{\sqrt{p}}\sqrt{\frac{p+q}{p-q}}=\frac{1}{\sqrt{p}}\sqrt{\frac{\left ( p-q \right )^{2}\left ( p+q \right )}{p-q}}=\frac{1}{\sqrt{p}}\sqrt{\left ( p-q \right )\left ( p+q \right )}=\frac{\sqrt{p^{2}-q^{2}}}{\sqrt{p}}\)

Ответ: \(\frac{\sqrt{p^{2}-q^{2}}}{\sqrt{p}}\)

Решение №16931: \(\left ( \frac{t\sqrt{t+2}}{\sqrt{t-2}}-\frac{2\sqrt{t-2}}{\sqrt{t+2}}-\frac{4t}{\sqrt{t^{2}-4}} \right )^{\frac{1}{2}}:\sqrt[4]{t^{2}-4}=\left ( \frac{t\left ( t+2 \right )-2\left ( t-2 \right )-4t}{\sqrt{t^{2}-4}} \right )^{\frac{1}{2}}:\sqrt[4]{t^{2}-4}=\left ( \frac{t^{2}-4t+4}{\sqrt{t^{2}-4}} \right )^{\frac{1}{2}}\cdot \frac{1}{\sqrt[4]{t^{2}-4}}=\frac{t-2}{\sqrt[4]{t^{2}-4}}\cdot \frac{1}{\sqrt[4]{t^{2}-4}}=\frac{t-2}{\sqrt{t^{2}-4}}=\frac{\sqrt{t-2}}{\sqrt{t+2}}=\frac{\sqrt{t^{2}-4}}{t+2}\)

Ответ: \(\frac{\sqrt{t^{2}-4}}{t+2}\)

Решение №16932: \(\frac{\sqrt{\sqrt{3}+2}\sqrt[4]{7-4\sqrt{3}}+\sqrt[3]{\sqrt{x}\left ( x+27 \right )-9x-27}}{\sqrt{x}-2-\sqrt{2-\sqrt{3}}\sqrt[4]{7+4\sqrt{3}}}=\frac{\sqrt{2+\sqrt{3}}\sqrt[4]{\left ( 2-\sqrt{3} \right )^{2}}+\sqrt{x}-3}{\sqrt{x}-2-\sqrt{2-\sqrt{3}}\sqrt[4]{\left ( 2+\sqrt{3} \right )^{2}}}=\frac{\sqrt{2+\sqrt{3}}\sqrt{2-\sqrt{3}}+\sqrt{x}-3}{\sqrt{x}-2-\sqrt{2-\sqrt{3}}\sqrt{2+\sqrt{3}}}=\frac{\sqrt{4-3}+\sqrt{x}-3}{\sqrt{x}-2-\sqrt{4-3}}=\frac{1+\sqrt{x}-3}{\sqrt{x}-2-1}=\frac{\sqrt{x}-2}{\sqrt{x}-3}\)

Ответ: \(\frac{\sqrt{x}-2}{\sqrt{x}-3}\)

Решение №16933: \(\left ( \frac{\sqrt[4]{x^{3}}-\sqrt[4]{x}}{1-\sqrt{x}}+\frac{1+\sqrt{x}}{\sqrt[4]{x}} \right )^{2}\cdot \left ( 1+\frac{2}{\sqrt{x}}+\frac{1}{x} \right )^{-\frac{1}{2}}=\left ( \frac{\sqrt[4]{x}\left ( \sqrt{x}-1 \right )}{1-\sqrt{x}}+\frac{1+\sqrt{x}}{\sqrt[4]{x}} \right )^{2}\cdot \left (\frac{x}{\left ( \sqrt{x}+1 \right )}\right )^{\frac{1}{2}}=\left ( -\sqrt[4]{x} +\frac{1+\sqrt{x}}{\sqrt[4]{x}}\right )^{2}\cdot \frac{\sqrt{x}}{\sqrt{x}+1}=\frac{1}{\sqrt{x}}\cdot \frac{\sqrt{x}}{\sqrt{x}+1}=\frac{1}{\sqrt{x}+1}=\frac{1-\sqrt{x}}{1-x}\)

Ответ: \(\frac{1-\sqrt{x}}{1-x}\)

Решение №16934: \(\frac{\left ( \sqrt[5]{a^{\frac{4}{3}}} \right )^{\frac{3}{2}}}{\left ( \sqrt[5]{a^{4}} \right )^{3}}:\frac{\left ( \sqrt{a\sqrt[3]{a^{2}b}} \right )^{4}}{\left ( \sqrt[4]{a\sqrt{b}} \right )^{6}}=\frac{\left ( a^{\frac{4}{15}} \right )^{\frac{3}{2}}}{\left ( a^{\frac{4}{5}} \right )^{3}}\cdot \frac{\left ( \sqrt{a\cdot a^{\frac{2}{3}}b^{\frac{1}{3}}} \right )^{4}}{\left ( \sqrt[4]{ab^{\frac{1}{2}}} \right )^{6}}=\frac{a^{\frac{2}{5}}}{a^{\frac{12}{5}}}\cdot \frac{\left ( a^{\frac{5}{6}}b^{\frac{1}{6}} \right )^{4}}{a^{\frac{6}{4}}b^{\frac{6}{8}}}=a^{-2}\cdot a^{\frac{10}{3}-\frac{3}{2}}\cdot b^{\frac{2}{3}-\frac{3}{4}}=a^{-2}\cdot a^{\frac{11}{6}}\cdot b^{-\frac{1}{12}}=\frac{1}{\sqrt[12]{a^{2}b}}\)

Ответ: \(\frac{1}{\sqrt[12]{a^{2}b}}\)

Решение №16935: \(\frac{\sqrt[3]{\sqrt{5}-\sqrt{3}}\sqrt[6]{8+2\sqrt{15}}-\sqrt[3]{a}}{\sqrt[3]{\sqrt{2}+\sqrt{12}}\sqrt[6]{8-2\sqrt{15}}-2\sqrt[3]{2a}+\sqrt[3]{a^{2}}}=\frac{\sqrt[3]{5-3}-\sqrt[3]{a}}{\sqrt[3]{2\left ( 5-3 \right )}-2\sqrt[3]{2a}+\sqrt[3]{a^{2}}}=\frac{\sqrt[3]{2}-\sqrt[3]{a}}{\left ( \sqrt[3]{2}-\sqrt[3]{a} \right )}=\frac{1}{\sqrt[3]{2}-\sqrt[3]{a}}\)

Ответ: \(\frac{1}{\sqrt[3]{2}-\sqrt[3]{a}}\)

Решение №16936: \(\frac{\sqrt{\sqrt{5}-2}\sqrt[4]{9+4\sqrt{5}}+\sqrt[3]{a^{2}}-\sqrt[3]{a}}{\sqrt{\sqrt{5}+2}\sqrt[4]{9-4\sqrt{5}}+a}=\frac{\sqrt{\sqrt{5}-2}\sqrt[4]{5+4\sqrt{5}+4}+\sqrt[3]{a^{2}}-\sqrt[3]{a}}{\sqrt{\sqrt{5}+2}\sqrt[4]{5-4\sqrt{5}+4}+a}=\frac{1+\sqrt[3]{a^{2}}-\sqrt[3]{a}}{1+a}=\frac{\sqrt[3]{a^{2}}-\sqrt[3]{a}+1}{\left ( \sqrt[3]{a} \right )^{3}+1}=\frac{1}{\sqrt[3]{a}+1}\)

Ответ: \(\frac{1}{\sqrt[3]{a}+1}\)

Решение №16937: \(\frac{1+2a^{\frac{1}{4}}-a^{\frac{1}{2}}}{1-a+4a^{\frac{3}{4}}-4a^{\frac{1}{2}}}+\frac{a^{\frac{1}{4}}-2}{\left ( a^{\frac{1}{4}}-1 \right )^{2}}=\frac{a^{\frac{2}{4}}-2a^{\frac{1}{4}}-1}{\left ( a^{\frac{1}{4}}-1\left ( a^{\frac{2}{4}}-a^{\frac{2}{4}} \right )-\left ( 2a^{\frac{2}{4}}-2a^{\frac{1}{4}} \right )-\left ( a^{\frac{1}{4}}-1 \right ) \right )}+\frac{a^{\frac{1}{4}}-2}{\left ( a^{\frac{1}{4}}-1 \right )^{2}}=\frac{1}{( a^{\frac{1}{4}}-1 \right )^{2}}+\frac{a^{\frac{1}{4}}-2}{\left ( a^{\frac{1}{4}}-1 \right )^{2}}=\frac{1+a^{\frac{1}{4}}-2}{\left ( a^{\frac{1}{4}}-1 \right )^{2}}=\frac{a^{\frac{1}{4}}-1}{\left ( a^{\frac{1}{4}}-1 \right )^{2}}=\frac{1}{\sqrt[4]{a}-1}\)

Ответ: \(\frac{1}{\sqrt[4]{a}-1}\)

Решение №16938: \(\frac{\sqrt{\sqrt{\frac{x-1}{x+1}}+\sqrt{\frac{x+1}{x-1}}-2}\left ( 2x+\sqrt{x^{2}-1} \right )}{\sqrt{\left ( x+1 \right )^{3}}-\sqrt{\left ( x-1 \right )^{3}}}=\frac{\sqrt{\frac{\left ( \sqrt{x+1}-\sqrt{x-1} \right )^{2}}{\sqrt{x^{2}-1}}}}{\sqrt{x+1}-\sqrt{x-1}}=\sqrt{\frac{\left ( \sqrt{x+1}-\sqrt{x-1} \right )^{2}}{\sqrt{x^{2}-1}\left ( \sqrt{x+1}-\sqrt{x-1} \right )^{2}}}=\sqrt{\frac{1}{\sqrt{x^{2}-1}}}=\frac{1}{\sqrt[4]{x^{2}-1}}\)

Ответ: \(\frac{1}{\sqrt[4]{x^{2}-1}}\)

Решение №16939: \(\left ( \frac{a+2}{\sqrt{2a}}-\frac{a}{\sqrt{2a}+2}+\frac{2}{a-\sqrt{2a}} \right )\cdot \frac{\sqrt{a}-\sqrt{2}}{a+2}=\left ( \frac{a+2}{\sqrt{2a}}-\frac{a}{\sqrt{2}\left ( \sqrt{a}+\sqrt{2} \right )}+\frac{2}{\sqrt{a}\left ( \sqrt{a}-\sqrt{2} \right )} \right )\cdot \frac{\sqrt{a}-\sqrt{2}}{a+2}=\frac{a^{2}-4-a^{2}+a\sqrt{2a}+2\sqrt{2a}+4}{\sqrt{2a}\left ( \sqrt{a}+\sqrt{2} \right )}\cdot \frac{1}{a+2}=\frac{\sqrt{2a}\left ( a+2 \right )}{\sqrt{2a}\left ( \sqrt{a}+\sqrt{2} \right )\left ( a+2 \right )}=\frac{1}{\sqrt{a}+\sqrt{2}}\)

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

Решение №16940: \(\left ( \left ( a-3\sqrt[6]{a^{5}}+9\sqrt[3]{a^{2}} \right )\left ( \sqrt{a} +3\sqrt[3]{a}+3\sqrt[12]{a^{5}}\right )^{-1}+3\sqrt[12]{a^{5}} \right )^{-1}=\left ( \frac{\sqrt[12]{a^{12}}-3\sqrt[12]{a^{10}}+9\sqrt[12]{a^{8}}}{\sqrt[12]{a^{6}}+3\sqrt[12]{a^{4}}+3\sqrt[12]{a^{5}}} +3\sqrt[12]{a^{5}}\right )^{-1}=\left ( \frac{\sqrt[12]{a^{4}}\left ( \sqrt[12]{a^{2}+3\sqrt[12]{a}+3} \right )\left ( \sqrt[12]{a^{2}}+3 \right )}{\sqrt[12]{a^{2}}+3\sqrt[12]{a}+3} \right )^{-1}=\left ( \sqrt[12]{a^{4}}\left ( \sqrt[12]{a^{2}}+3 \right ) \right )^{-1}=\frac{1}{\sqrt[12]{a^{4}}\left ( \sqrt[12]{a^{2}}+3 \right )}=\frac{1}{\sqrt[12]{a^{6}}+3\sqrt[12]{a^{4}}}=\frac{1}{\sqrt{a}+3\sqrt[3]{a}}\)

Ответ: \(\frac{1}{\sqrt{a}+3\sqrt[3]{a}}\)

Решение №16941: \(\frac{\sqrt{\frac{1}{a+2\sqrt{a-2}-1}}+\sqrt{\frac{1}{a-2\sqrt{a-2}-1}}}{\sqrt{\frac{1}{a+2\sqrt{a-2}-1}}-\sqrt{\frac{1}{a-2\sqrt{a-2}-1}}}=\frac{\sqrt{\frac{1}{\left ( \sqrt{a-2}+1 \right )^{2}}}+\frac{1}{\left ( \sqrt{a-2}-1 \right )^{2}}}{\frac{1}{\left ( \sqrt{a-2}+1 \right )^{2}}-\frac{1}{\left ( \sqrt{a-2}-1 \right )^{2}}}=\frac{\left | \sqrt{a-2}-1 \right |+\sqrt{a-2}+1}{\left | \sqrt{a-2}-1 \right |-\sqrt{a-2}-1}=-\frac{1}{\sqrt{a-2}};-\sqrt{a-2}\)

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

Решение №16942: \(\frac{\sqrt{2b+2\sqrt{b^{2}-4}}}{\sqrt{b^{2}-4}+b+2}=\frac{\sqrt{b+2\sqrt{\left ( b+2 \right )\left ( b-2 \right )}}+b}{\sqrt{b^{2}-4}+b+2}=\frac{\sqrt{b+2}+\sqrt{b-2}}{\sqrt{b+2}\left ( \sqrt{b-2}+\sqrt{b+2} \right )}=\frac{1}{\sqrt{b+2}}\)

Ответ: \(\frac{1}{\sqrt{b+2}}\)

Решение №16943: \(\frac{4x\left ( x+\sqrt{x^{2}-1} \right )^{2}}{\left ( x+\sqrt{x^{2}-1} \right )^{4}-1}=\frac{4x\left ( x+\sqrt{x^{2}-1} \right )^{2}}{\left ( x^{2}+2x\sqrt{x^{2}-1}+x^{2}-1 \right )^{2}-1}=\frac{4x\left ( x+\sqrt{x^{2}-1} \right )^{2}}{\left ( 2x^{2}+2x\sqrt{x^{2}-1}-1 \right )^{2}-1}=\frac{4x\left ( x+\sqrt{x^{2}-1} \right )^{2}}{\left ( 2x^{2}+2x\sqrt{x^{2}-1}-2 \right )\left ( 2x^{2}+2x\sqrt{x^{2} \right )}=\frac{x+\sqrt{x^{2}-1}}{\left ( x^{2}-1 \right )+x\sqrt{x^{2}-1}}=\frac{1}{\sqrt{x^{2}-1}}\)

Ответ: \(\frac{1}{\sqrt{x^{2}-1}}\)

Решение №16944: \(\frac{\left | x-1 \right |+\left | x \right |+x}{3x^{2}-4x+1}=\frac{-x+1-x+x}{\left ( x-1 \right )\left ( 3x-1 \right )};\frac{-x+1+x+x}{\left ( x-1 \right )\left ( 3x-1 \right )};\frac{x-1-x+x}{\left ( x-1 \right )\left ( 3x-1 \right )}=\frac{x-1}{\left ( x-1 \right )\left ( 3x-1 \right )};\frac{x+1}{\left ( x-1 \right )\left ( 3x-1 \right )};\frac{3x-1}{\left ( x-1 \right )\left ( 3x-1 \right )}=\frac{1}{1-3x};\frac{x+1}{\left ( x-1 \right )\left ( 3x-1 \right )};\frac{1}{x-1};\)

Ответ: \(\frac{1}{1-3x};\frac{x+1}{\left ( x-1 \right )\left ( 3x-1 \right )};\frac{1}{x-1};\)

Решение №16945: \(\frac{\left ( \sqrt[4]{a}+\sqrt[4]{b}-\sqrt[8]{ab} \right )\left ( \sqrt[4]{b}+\sqrt[4]{a}+\sqrt[8]{ab} \right )}{\sqrt[4]{a^{3}b}-b}:\frac{\left ( \sqrt[8]{a}+\sqrt[8]{b} \right )+\left ( \sqrt[8]{a}-\sqrt[8]{b} \right )^{2}}{\left ( \sqrt{a}-\sqrt{b} \right )b^{-\frac{1}{4}}}=\frac{\left ( \sqrt[8]{a^{2}}+\sqrt[8]{b^{2}} \right )^{2}-\left ( \sqrt[8]{ab} \right )^{2}}{\sqrt[8]{b^{2}\left ( \sqrt[8]{a^{2}}-\sqrt[8]{b^{2}} \right )}}:\frac{\left ( 2\sqrt[8]{a^{2}}+2\sqrt[8]{b^{2}} \right )\sqrt[8]{b^{2}}}{\sqrt[8]{a^{4}}-\sqrt[8]{b^{4}}}=\frac{1}{\sqrt[8]{b^{2}}\left ( \sqrt[8]{a^{2}}-\sqrt[8]{b^{2}} \right )}\cdot \frac{\sqrt[8]{a^{2}}-\sqrt[8]{b^{2}} }{2\sqrt[8]{b^{2}}}=\frac{1}{2\sqrt[8]{b^{4}}}=\frac{1}{2\sqrt{b}}\)

Ответ: \(\frac{1}{2\sqrt{b}}\)

Решение №16946: \(\frac{\sqrt{1+\left ( \frac{x^{2}-1}{2x} \right )^{2}}}{\left ( x^{2}+1 \right )\frac{1}{x}}=\frac{\sqrt{\frac{x^{4}+2x^{2}+1}{4x^{2}}}}{\frac{x^{2}+1}{x}}=\frac{\frac{x^{2}+1}{2\left | x \right |}}{\frac{x^{2}+1}{x}}=\frac{x}{2\left | x \right |}=-\frac{1}{2};\frac{1}{2}\)

Ответ: \(-\frac{1}{2};\frac{1}{2}\)

Решение №16947: \(\left ( \frac{4}{a+\frac{1}{b+\frac{1}{c}}}:\frac{1}{a+\frac{1}{b}}-\frac{4}{b\left ( abc+a+c \right )} \right )^{-\frac{1}{2}}=\left ( \frac{4}{a+\frac{c}{bc+1}}:\frac{b}{ab+1}-\frac{4}{b\left ( abc+a+c \right )} \right )^{-\frac{1}{2}}=\left ( \frac{4bc+4}{abc+a+c}\cdot \frac{ab+1}{b}-\frac{4}{b\left ( abc+a+c \right )} \right )^{-\frac{1}{2}}=\left ( \frac{4ab^{2}c+4bc+4ab+4}{b\left ( abc+a+c \right )} -\frac{4}{b\left ( abc+a+c \right )}\right )^{-\frac{1}{2}}=\left ( \frac{4ab^{2}c+4bc+4ab+4-4}{b\left ( abc+a+c \right )} \right )^{-\frac{1}{2}}=4^{-\frac{1}{2}}=\frac{1}{2}\)

Ответ: \(\frac{1}{2}\)

Решение №16948: \(\left ( \frac{2}{\sqrt{3}-1}+\frac{3}{\sqrt{3}-2}+\frac{15}{3-\sqrt{3}} \right )\left ( \sqrt{3}+5 \right )^{-1}=\left ( \frac{2\left ( \sqrt{3}+1 \right )}{2}+\frac{3\left ( \sqrt{3}+2 \right )}{-1}+\frac{15\left ( 3+\sqrt{3} \right )}{6} \right )\cdot \frac{1}{\sqrt{3}+5}=\frac{-4\sqrt{3}-10+15+5\sqrt{3}}{2}\cdot \frac{1}{\sqrt{3}+5}=\frac{\sqrt{3}+5}{2}\cdot \frac{1}{\sqrt{3}+5}=\frac{1}{2}\)

Ответ: \(\frac{1}{2}\)

Решение №16949: \(\frac{\sqrt{\sqrt[4]{8}-\sqrt{\sqrt{2}+1}}}{\sqrt{\sqrt[4]{8}+\sqrt{\sqrt{2}-1}}-\sqrt{\sqrt[4]{8}-\sqrt{\sqrt{2}-1}}}=\frac{1}{\sqrt{2}};\frac{\sqrt[4]{8}-\sqrt{\sqrt{2}+1}}{2\sqrt[4]{8}-2\sqrt{\sqrt{8}-\sqrt{2}+1}}=\frac{1}{2};\frac{\sqrt[4]{8}-\sqrt{\sqrt{2}+1}}{2\left ( \sqrt[4]{8}-\sqrt{\sqrt{2}+1}\right )}=\frac{1}{2};\frac{1}{2}=\frac{1}{2}\)

Ответ: \(\frac{1}{2}=\frac{1}{2}\\)

Решение №16950: \(\frac{\sqrt{x^{2}y^{-2}-xy^{-1}+\frac{1}{4}}\left ( xy^{-2}+y^{-\frac{3}{2}} \right )}{2x^{2}-y^{\frac{3}{2}}-xy+2xy^{\frac{1}{2}}}=\frac{\frac{\sqrt{\left ( 2x-y \right )^{2}}}{2y}\cdot \frac{x+\sqrt{y}}{y^{2}}}{\left ( x+\sqrt{y} \right )\left ( 2x-y \right )}=\frac{\left | 2x-y \right |}{2y^{3}\left ( 2x-y \right )}=-\frac{1}{2y^{3}};\frac{1}{2y^{3}}\)

Ответ: \(-\frac{1}{2y^{3}};\frac{1}{2y^{3}}\)

Решение №16951: \(\frac{\sqrt{c-d}}{c^{2}\sqrt{2c}}\left ( \sqrt{\frac{c-d}{c+d}}+\sqrt{\frac{c^{2}+cd}{c^{2}-cd}} \right )=\frac{\sqrt{c-d}}{c^{2}\sqrt{2c}}\left (\sqrt{\frac{c-d}{c+d}}+\sqrt{\frac{c\left ( c+d \right )}{c\left ( c-d \right )}} \right )=\frac{\sqrt{c-d}}{c^{2}\sqrt{2c}}\left ( \sqrt{\frac{c-d}{c+d}}+\frac{\sqrt{c+d}}{\sqrt{c-d}} \right )=\frac{2c}{c^{2}\sqrt{2c}\sqrt{c+d}}=\frac{\sqrt{2}}{c\sqrt{c}\sqrt{c+d}}=\frac{\sqrt{2}}{c\sqrt{c^{2}+cd}}=\frac{\sqrt{2}}{2\sqrt{4+2\cdot \frac{1}{4}}}=\frac{1}{\sqrt{9}}=\frac{1}{3}\)

Ответ: \(\frac{1}{3}\)

Решение №16952: \(\frac{a^{-1}-b^{-1}}{a^{-3}+b^{-3}}:\frac{a^{2}b^{2}}{\left ( a+b \right )^{2}-3ab}\cdot \left ( \frac{a^{2}-b^{2}}{ab} \right )^{-1}=\frac{\frac{1}{a}-\frac{1}{b}}{\frac{1}{a^{3}}-\frac{1}{b^{3}}}:\frac{a^{2}b^{2}}{a^{2}+2ab+b^{2}-3ab}\cdot \frac{ab}{a^{2}-b^{2}}=-\frac{ab}{\left ( a+b \right )^{2}}=-\frac{\left ( 1-\sqrt{2} \right )\left ( 1+\sqrt{2} \right )}{\left ( 1-\sqrt{2}+1+\sqrt{2} \right )^{2}}=\frac{1}{4}\)

Ответ: \(\frac{1}{4}\)

Решение №16953: \(\frac{x\left ( x+1 \right )\left ( x+2 \right )\left ( x+3 \right )}{\left ( x-1 \right )\left ( x+4 \right )}; x=\frac{\sqrt{5}-3}{2};=\frac{\frac{\sqrt{5}-3}{2}\left (\frac{\sqrt{5}-3}{2}+1 \right )\left (\frac{\sqrt{5}-3}{2}+2 \right )\left ( \frac{\sqrt{5}-3}{2}+3 \right )}{\left ( \frac{\sqrt{5}-3}{2}-1 \right )\left ( \frac{\sqrt{5}-3}{2}+4 \right )}=\frac{\left ( \left ( \frac{\sqrt{5}-3}{2} \right )^{2}+3\frac{\sqrt{5}-3}{2} \right )\left ( \left ( \frac{\sqrt{5}-3}{2} \right )^{2}+3\frac{\sqrt{5}-3}{2}+2 \right )}{\left ( \frac{\sqrt{5}-3}{2}-1 \right )^{2}+3\frac{\sqrt{5}-3}{2}-4}=\frac{\left ( \frac{7-3\sqrt{5}+3\sqrt{5}-9}{2} \right )^{2}+2\left (\frac{7-3\sqrt{5}+3\sqrt{5}-9}{2} \right )}{-1-4}=\frac{1}{5}\)

Ответ: \(\frac{1}{5}\)

Решение №16954: \(\frac{\left ( a^{\frac{1}{m}}-a^{\frac{1}{n}} \right )^{2}+4a^{\frac{m+n}{mn}}}{\left ( a^{\frac{2}{m}}-a^{\frac{2}{n}} \right )\left ( \sqrt[m]{a^{m+1}}+\sqrt[n]{a^{n+1}} \right )}=\frac{a^{\frac{2}{m}}-2a^{\frac{1}{m}+\frac{1}{n}}+a^{\frac{2}{n}}+4a^{\frac{1}{m}+\frac{1}{n}}}{\left ( a^{\frac{1}{m}}-a\frac{1}{n} \right )\left ( a^{\frac{1}{m}}+a^{\frac{1}{n}} \right )\left ( a^{1+\frac{1}{m}}+a^{1+\frac{1}{n}} \right )}=\frac{\left ( a^{\frac{1}{m}}+a^{\frac{1}{n}} \right )^{2}}{\left ( a^{\frac{1}{m}}-a\frac{1}{n} \right )\left ( a^{\frac{1}{m}}+a^{\frac{1}{n}} \right )a\left ( a^{\frac{1}{m}}+a^{\frac{1}{n}} \right )}=\frac{1}{a\left ( a^{\frac{1}{m}}-a^{\frac{1}{n}} \right )}=\frac{1}{a\left ( \sqrt[m]{a}-\sqrt[n]{a} \right )}\)

Ответ: \(\frac{1}{a\left ( \sqrt[m]{a}-\sqrt[n]{a} \right )}\)

Решение №16955: \(\frac{b^{-\frac{1}{6}}\sqrt{a^{3}b}\sqrt[3]{a^{3}b}-\sqrt{a^{3}b^{2}}\sqrt[3]{b^{2}}}{\left ( 2a^{2}-b^{2}-ab \right )\sqrt[6]{a^{9}b^{4}}}:\left ( \frac{3a^{3}}{2a^{2}-ab-b^{2}}-\frac{ab}{a-b} \right )=\frac{\sqrt[6]{a^{15}b^{5}}-\sqrt[6]{a^{9}b^{10}}}{\left ( a-b \right )\left ( 2a+b \right )\sqrt[6]{a^{9}b^{4}}}:\frac{3a^{3}-ab\left ( 2a+b \right )}{\left ( a-b \right )\left ( 2a+b \right )}=\frac{1}{2a+b}\cdot \frac{\left ( a-b \right )2a+b}{3a^{3}-2a^{2}b-ab^{2}}=\frac{a-b}{\left ( a-b \right )\left ( 2a^{2}+a\left ( a+b \right ) \right )}=\frac{1}{3a^{2}+ab}=\frac{1}{a\left ( 3a+b \right )}\)

Ответ: \(\frac{1}{a\left ( 3a+b \right )}\)

Решение №16956: \(\frac{1}{2\left ( 1+\sqrt{a} \right )}+\frac{1}{2\left ( 1-\sqrt{a} \right )}-\frac{a^{2}+2}{1-a^{3}}=\frac{1-\sqrt{a}+1+\sqrt{a}}{2\left ( 1+\sqrt{a} \right )\left ( 1-\sqrt{a} \right )}-\frac{a^{2}+2}{1-a^{3}}=\frac{2}{2\left ( 1-a \right )}-\frac{a^{2}+2}{\left ( 1-a \right )\left ( 1+a+a^{2} \right )}=\frac{1}{1-a}-\frac{a^{2}+2}{\left ( 1-a \right )\left ( 1+a+a^{2} \right )}=\frac{1+a+a^{2}-a^{2}-2}{\left ( 1-a \right )\left ( 1+a+a^{2} \right )}=\frac{-1}{a^{2}+a+1}\)

Ответ: \(\frac{-1}{a^{2}+a+1}\)