Решение №16897: \(\left ( \frac{3^{\frac{3}{2}}+\frac{1}{8}z^{\frac{3}{5}}}{3+\sqrt{3}\sqrt[5]{z}+\frac{1}{4}\sqrt[5]{z^{2}}}+\frac{3\sqrt{3}\sqrt[5]{z}}{2\sqrt{3}+\sqrt[5]{z}} \right )^{-1}:\frac{1}{2\sqrt{12}+\sqrt[5]{32z}}=\left ( \frac{24\sqrt{3}+\sqrt[5]{z^{3}}}{24+8\sqrt{3}\sqrt[5]{z}+2\sqrt[5]{z^{2}}}+\frac{3\sqrt{3}\sqrt[5]{z}}{2\sqrt{3}+\sqrt[5]{z}} \right )^{-1}:\frac{1}{2\left ( 2\sqrt{3}+\sqrt[5]{z} \right )}=\left ( \frac{\left ( 2\sqrt{3}+\sqrt[5]{z} \right )^{2}}{2\left ( 2\sqrt{3}+\sqrt[5]{z} \right )} \right )^{-1}\cdot 2\left ( 2\sqrt{3}+\sqrt[5]{z}\right ) =\frac{2}{2\sqrt{3}+\sqrt[5]{z}}\cdot 2\left ( 2\sqrt{3}+\sqrt[5]{z} \right )=4\)

Ответ: 4

Решение №16898: \(\frac{\sqrt{\frac{abc+4}{a}+4\sqrt{\frac{bc}{a}}}}{\sqrt{abc}+2}=\frac{\sqrt{\frac{abc+4}{a}+\frac{4\sqrt{bc}}{\sqrt{a}}}}{\sqrt{abc}+2}=\frac{\sqrt{\frac{abc+4\sqrt{abc}+4}{a}}}{\sqrt{abc}+2}=\frac{\sqrt{\frac{\left ( \sqrt{abc}+2 \right )^{2}}{a}}}{\sqrt{abc}+2}=\frac{1}{\sqrt{a}}=\frac{1}{\sqrt{0.04}}=\frac{1}{0.2}=5\)

Ответ: 5

Решение №16899: \(\frac{\left ( a^{2}-b^{2} \right )\left ( a^{2}+\sqrt[3]{b^{2}}+a\sqrt[3]{b} \right )}{a\sqrt[3]{b}+a\sqrt{a}-b\sqrt[3]{b}-\sqrt{ab^{2}}}:\frac{a^{3}-b}{a\sqrt[3]{b}-\sqrt[6]{a^{3}b^{2}}-\sqrt[3]{b^{2}}-a\sqrt{a}}=\frac{\left ( a-b \right )\left ( a+b \right )\left ( a^{2}+a\sqrt[3]{b}+\sqrt[3]{b^{2}} \right )}{a\left ( \sqrt{a}+\sqrt[3]{b} \right )-b\left ( \sqrt{a}+\sqrt[3]{b} \right )}:\frac{a^{3}-b}{a\left ( \sqrt{a}+\sqrt[3]{b} \right )-\sqrt[3]{b}\left ( \sqrt{a}+\sqrt[3]{b} \right )}=\frac{\left ( a+b \right )\left ( a^{2}+a\sqrt[3]{b}+\sqrt[3]{b^{2}} \right )}{\sqrt{a}+\sqrt[3]{b}}\cdot \frac{\left ( \sqrt{a}+\sqrt[3]{b} \right )\left ( a-\sqrt[3]{b} \right )}{\left ( a-\sqrt[3]{b} \right )\left ( a^{2}+a\sqrt[3]{b}+\sqrt[3]{b^{2}} \right )}=a+b=4.91+0.09=5\)

Ответ: 5

Решение №16900: \(\left ( \sqrt{25-x^{2}}-\sqrt{15-x^{2}} \right )\left ( \sqrt{25-x^{2}}+\sqrt{15-x^{2}} \right )=2\left ( \sqrt{25-x^{2}}+\sqrt{15-x^{2}} \right ) 25-x^{2}-15+x^{2}=2\left (\sqrt{25-x^{2}}+\sqrt{15-x^{2}} \right ), \sqrt{25-x^{2}}+\sqrt{15-x^{2}}=5\)

Ответ: 5

Решение №16901: \(\left ( x^{4}-7x^{2}+1 \right )^{-2}\cdot \left ( \left ( x^{2}+\frac{1}{x^{2}} \right )^{2}-14\left ( x+\frac{1}{x} \right )^{2}+77 \right )^{\frac{1}{2}}=\frac{1}{\left ( x^{4}-7x^{2}+1 \right )^{2}}\cdot \left ( \left ( x+\frac{1}{x} \right )^{4} -18\left ( x+\frac{1}{x} \right )^{2}+81\right )=\frac{1}{\left ( x^{4}-7x^{2}+1 \right )^{2}}\cdot \sqrt{\left ( \left ( x+\frac{1}{x} \right )^{2}-9 \right )^{2}}=\frac{1}{\left ( x^{4}-7x^{2}+1 \right )^{2}}\cdot \frac{\left ( x^{4}-7x^{2}+1 \right )^{2}}{x^{4}}=\frac{1}{x^{4}}=\frac{1}{\left ( \frac{\sqrt[4]{125}}{5} \right )^{4}}=5\)

Ответ: 5

Решение №16902: \(\left ( \frac{9}{a+8}-\frac{a^{\frac{1}{3}}+2}{a^{\frac{2}{3}}-2a^{\frac{1}{3}}+4} \right )\frac{a^{\frac{4}{3}}+8a^{\frac{1}{3}}}{1-a^{\frac{2}{3}}}+\frac{5-a^{\frac{2}{3}}}{1+a^{\frac{1}{3}}}=\frac{\left ( 3-a^{\frac{1}{3}}-2 \right )\left ( 3+a^{\frac{1}{3}}+2 \right )}{a+8}\cdot \frac{a^{\frac{1}{3}}\left ( a+8 \right )}{\left ( 1-a^{\frac{1}{3}} \right )\left ( 1+a^{\frac{1}{3}} \right )}+\frac{5-a^{\frac{2}{3}}}{1+a^{\frac{1}{3}}}=\frac{5a^{\frac{1}{3}}+a^{\frac{2}{3}}+5-a^{\frac{2}{3}}}{1+a^{\frac{1}{3}}}=\frac{5a^{\frac{1}{3}}+5}{1+a^{\frac{1}{3}}}=5\)

Ответ: 5

Решение №16903: \(\frac{3\sqrt{12}}{\sqrt{45}-4\sqrt{3}}+5\sqrt{2.4}\left ( \sqrt{15}+3 \right )=\frac{3\sqrt{3\cdot 4}}{\sqrt{3\cdot 15}-4\sqrt{3}}+5\sqrt{\frac{12}{5}}\left ( \sqrt{15}+3 \right )=\frac{6}{\sqrt{15}-4}+30+6\sqrt{15}=-6\sqrt{15}-24+30+6\sqrt{15}=6\)

Ответ: 6

Решение №16904: \(\\frac{\left ( 1-y \right )\left ( y+2 \right )}{y^{2}\left ( y+1 \right )^{2}}; y=\frac{\sqrt{3}-1}{2};=\frac{\left (1-\frac{\sqrt{3}-1}{2} \right )\left (\frac{\sqrt{3}-1}{2}+2 \right )}{\left ( \frac{\sqrt{3}-1}{2} \right )^{2}\left ( \frac{\sqrt{3}-1}{2}+1 \right )^{2}}=\frac{\left (\frac{\sqrt{3}-1}{2} \right )^{2}+\frac{\sqrt{3}-1}{2}-2 }{\left (\left ( \frac{\sqrt{3}-1}{2} \right )^{2}+\frac{\sqrt{3}-1}{2} \right )}=\frac{\frac{2-\sqrt{3}}{2}+\frac{\sqrt{3}-1}{2}-2}{\left (\frac{2-\sqrt{3}}{2}+\frac{\sqrt{3}} \right )^{2}}=-\frac{\frac{2-\sqrt{3}+\sqrt{3}-1}{2}-2}{\left ( \frac{2-\sqrt{3}+\sqrt{3}-1}{2} \right )^{2}}=6)

Ответ: 6

Решение №16905: \(8-a+2\sqrt{\left ( 8-a \right )\left ( 5+a \right )}+5+a=25; \sqrt{\left ( 8-a \right )\left ( 5+a \right )}=6\)

Ответ: 6

Решение №16906: \(\frac{\left ( x^{2}-y^{2} \right )\left ( \sqrt[3]{x}+\sqrt[3]{y} \right )}{\sqrt[3]{x^{5}}+\sqrt[3]{x^{2}y^{3}}-\sqrt[3]{x^{2}y^{2}}-\sqrt[3]{y^{5}}}-\left ( \sqrt[3]{xy}+\sqrt[3]{y^{2}} \right )=\frac{\left ( \sqrt[3]{x}-\sqrt[3]{y} \right )\left ( \sqrt[3]{x^{2}}+\sqrt[3]{xy}+\sqrt[3]{y^{2}} \right )}{\sqrt[3]{x}-\sqrt[3]{y}}-\sqrt[3]{xy}-\sqrt[3]{y^{2}}=\sqrt[3]{x^{2}}+\sqrt[3]{xy}+\sqrt[3]{y^{2}}-\sqrt[3]{xy}-\sqrt[3]{y^{2}}=\sqrt[3]{x^{2}}=\sqrt[3]{64^{2}}=4^{2}=16\)

Ответ: 16

Решение №16907: \(\left ( -4a\sqrt[3]{\frac{\sqrt{ax}}{a^{2}}} \right )^{3}+\left ( -10a\sqrt{x}\sqrt{\left ( ax \right )^{-1}} \right )^{2}+\left ( -2\left ( \sqrt[3]{a\sqrt[4]{\frac{x}{a}}} \right )^{2} \right )^{3}=\frac{-64a^{3}\sqrt{ax}}{a^{2}}+\frac{100a^{2}x}{ax}-\frac{8a^{2}\sqrt{x}}{\sqrt{a}}=-64a\sqrt{ax}+100a-8a\sqrt{ax}=100a-72a\sqrt{ax}=100\cdot 3\frac{4}{7}-72\cdot 3\frac{4}{7}\cdot \sqrt{3\frac{4}{7}\cdot 0.28}=100\cdot \frac{25}{7}-72\cdot \frac{25}{7}\cdot \sqrt{\frac{25}{7}\cdot \frac{7}{25}}=\frac{2500}{7}-\frac{1800}{7}=\frac{700}{7}=100\)

Ответ: 100

Решение №16908: \(a+b=11, ab=21; a^{3}+b^{3}=\left ( a+b \right )\left ( a^{2}-ab+b^{2} \right )=\left ( a+b \right )\left ( \left ( a+b \right )^{2}-3ab \right )=11\left ( 11^{2}-3*21 \right )=11\left ( 121-63 \right )=638\)

Ответ: 638

Решение №16909: \(a^{2}c^{2}+2abcd+b^{2}d^{2}+a^{2}d^{2}-2abcd+b^{2}c^{2}=\left ( ac+bd \right )^{2}+\left ( ad-bc \right )^{2} \left ( a^{2}+b^{2} \right )\left ( c^{2}+d^{2} \right )=\left ( ac+bd \right )^{2}+\left ( ad-bc \right )^{2}\)

Ответ: -

Решение №16910: \(\frac{a-1}{\sqrt{a}-\sqrt[3]{a}}=\frac{\left ( a-1 \right )\left ( \sqrt[6]{a^{15}}+\sqrt[6]{a^{12}*2^{2}}+\sqrt[6]{a^{9}*a^{4}}+\sqrt[6]{a^{6}*a^{6}}+\sqrt[6]{a^{3}*a^{8}}+\sqrt[6]{a^{10}} \right )}{\left ( \sqrt[6]{a^{3}}-\sqrt[6]{a^{2}} \right )\left ( \sqrt[6]{a^{15}}+\sqrt[6]{a^{12}*a^{2}}+\sqrt[6]{a^{9}*a^{4}}+\sqrt[6]{a^{6}*a^{6}}+\sqrt[6]{a^{3}*a^{8}}+\sqrt[6]{a^{10}}+\sqrt[8]\right )}=\frac{\left ( \sqrt[6]{a}+1 \right )\sqrt[6]{a^{2}}\left ( \sqrt[6]{a^{6}}+\sqrt[6]{a^{4}}+\sqrt[6]{a^{2}} \right )}{a}=\frac{\left ( \sqrt{a}+\sqrt[3]{a} \right )\left ( a+\sqrt[3]{2}+\sqrt[3]{a} \right )}{a}\)

Ответ: \(\frac{\left ( \sqrt{a}+\sqrt[3]{a} \right )\left ( a+\sqrt[3]{2}+\sqrt[3]{a} \right )}{a}\)

Решение №16911: \(\frac{\left ( \sqrt{a^{2}+a\sqrt{a^{2}-b^{2}}} \right )-\left ( \sqrt{a^{2}-a\sqrt{a^{2}-b^{2}}} \right )^{2}}{2\sqrt{a^{3}b}}\cdot \left ( \sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}-2 \right ) Х=\frac{\left ( \sqrt{a^{2}+a\sqrt{a^{2}-b^{2}}} \right )-\left ( \sqrt{a^{2}-a\sqrt{a^{2}-b^{2}}} \right )^{2}}{2\sqrt{a^{3}b}}=\frac{\left ( \sqrt{a\left ( a+\sqrt{a^{2}-b^{2}} \right )-\sqrt{a\left ( a-\sqrt{a^{2}-b^{2}}\right} \right )^{2}}{2a\sqrt{ab}}=\frac{a\left ( a+\sqrt{a^{2}-b^{2}}-2\sqrt{a^{2}-a^{2}+b^{2}}+a-\sqrt{a^{2}-b^{2}} \right )}{2a\sqrt{ab}}=\frac{2a-2b}{2a\sqrt{ab}}=\frac{a-b}{\sqrt{ab}} Y=\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}-2=\frac{\sqrt{a}}{\sqrt{b}+\frac{\sqrt{b}}{\sqrt{a}}-2}=\frac{\left ( \sqrt{a}\right )^{2}-2\sqrt{ab}+\left ( \sqrt{b} \right )^{2}}{\sqrt{ab }}=\frac{\left ( \sqrt{a}-\sqrt{b} \right )^{2}}{\sqrt{ab}} X:Y=\frac{a-b}{\sqrt{ab}}:\frac{\left ( \sqrt{a}-\sqrt{b} \right )^{2}}{\sqrt{ab}}=\frac{\left ( a-b \right )\sqrt{ab}}{\sqrt{ab}\left ( \sqrt{a}-\sqrt{b} \right )^{2}}=\frac{a-b}{\left ( \sqrt{a}-\sqrt{b} \right )^{2}}= \frac{\left ( \sqrt{a}+\sqrt{b} \right )^{2}}{a-b}\)

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

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