Решение №17094: \(\frac{14}{\sqrt[4]3{+\sqrt[8]{2}}}=\frac{14}{\sqrt[8]{9}+\sqrt[8]{2}}=\frac{14\left ( \sqrt[8]{9^{7}}-\sqrt[8]{9^{6}*2}-\sqrt[8]{9^{4}*2^{2}}-\sqrt[8]{9^{4}*2^{3}}+\sqrt[8]{9^{3}*2^{4}}-\sqrt[8]{9^{2}*2^{5}}+\sqrt[8]{9*2^{6}}-\sqrt[8]{2^{7}} \right )}{\left ( \sqrt[8]{9}+\sqrt[8]{2} \right )\left ( \sqrt[8]{9^{7}}-\sqrt[8]{9^{6}*2}-\sqrt[8]{9^{4}*2^{2}}-\sqrt[8]{9^{4}*2^{3}}+\sqrt[8]{9^{3}*2^{4}}-\sqrt[8]{9^{2}*2^{5}}+\sqrt[8]{9*2^{6}}-\sqrt[8]{2^{7}} \right )}=\frac{14\left ( \sqrt[8]{9^{3}}-\sqrt[8]{9^{2}*2}+\sqrt[8]{9*2^{2}}-\sqrt[8]{2^{3}} \right )\left ( \sqrt[8]{9^{4}}+\sqrt[8]{2^{4}} \right )}{9-2}=2\left ( \sqrt[4]{3}-\sqrt[8]{2} \right )\left ( \sqrt{3}+\sqrt[4]{2} \right )\left ( 3+\sqrt{2} \right )\)

Ответ: \(2\left ( \sqrt[4]{3}-\sqrt[8]{2} \right )\left ( \sqrt{3}+\sqrt[4]{2} \right )\left ( 3+\sqrt{2} \right )\)

Решение №17095: \(\frac{\left ( \sqrt[8]{x}+\sqrt[8]{y} \right )^{2}+\left ( \sqrt[8]{x}-\sqrt[8]{y} \right )^{2}}{x-\sqrt{xy}}:\frac{\left ( \sqrt[4]{x}+\sqrt[8]{xy}+\sqrt[4]{y} \right )^{2}\left ( \sqrt[4]{x}-\sqrt[8]{xy}+\sqrt[4]{y} \right )^{2}}{\sqrt[4]{x^{3}y}-y}=\frac{2}{\sqrt[8]{x^{4}}\left ( \sqrt[8]{x^{2}}-\sqrt[8]{y^{2}} \right )}:\frac{\sqrt[8]{x^{4}}+\sqrt[8]{x^{2}y^{2}}+\sqrt[8]{y^{4}}}{\sqrt[8]{y^{2}}\left ( \sqrt[8]{x^{2}}-\sqrt[8]{y^{2}} \right )\left (\sqrt[8]{x^{4}}+\sqrt[8]{x^{2}y^{2}}+\sqrt[8]{y^{4}} \right )}=\frac{2}{\sqrt[8]{x^{4}}\left ( \sqrt[8]{x^{2}}-\sqrt[8]{y^{2}} \right )}:\frac{1}{\sqrt[8]{y^{2}}\left ( \sqrt[8]{x^{2}}-\sqrt[8]{y^{2}} \right )}=\frac{2\sqrt[8]{y^{2}}\left ( \sqrt[8]{x^{2}}-\sqrt[8]{y^{2}} \right )}{\sqrt[8]{x^{4}}\left ( \sqrt[8]{x^{2}}-\sqrt[8]{y^{2}} \right )}=\frac{2\sqrt[8]{y^{2}}}{\sqrt[8]{x^{4}}}=2\sqrt[4]{\frac{y}{x^{2}}}\)

Ответ: \(2\sqrt[4]{\frac{y}{x^{2}}}\)

Решение №17096: \(5\sqrt{48\sqrt[3]{\frac{2}{3}}}+\sqrt{32\sqrt[3]{\frac{9}{4}}}-11\sqrt[3]{12\sqrt{8}}=5\sqrt{16\cdot 3\sqrt[3]{\frac{2}{3}}}+\sqrt{16\cdot 2\sqrt[3]{\frac{9}{4}}}-11\sqrt[3]{12\cdot 2\sqrt{2}}=20\sqrt{\sqrt[3]{18}}+4\sqrt{\sqrt[3]{18}}-22\sqrt[3]{\sqrt{9\cdot 2}}=24\sqrt{\sqrt[3]{18}}-22\sqrt{\sqrt[3]{18}}=2\sqrt[6]{18}\)

Ответ: \(2\sqrt[6]{18}\)

Решение №17097: \(\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}}=\sqrt{x-2+2\sqrt{2\left ( x-2 \right )}+2}+\sqrt{x-2-2\sqrt{2\left ( x-2 \right )}+2}=\sqrt{\left ( \sqrt{x-2}+\sqrt{2} \right )^{2}}+\sqrt{\left ( \sqrt{x-2}-\sqrt{2} \right )^{2}}=\sqrt{x-2}+\sqrt{2}+\left | \sqrt{x-2}-\sqrt{2} \right |=2\sqrt{2};2\sqrt{x-2}\)

Ответ: \(2\sqrt{2};2\sqrt{x-2}\)

Решение №17098: \(\sqrt[3]{45+29\sqrt{2}}-\sqrt[3]{45-29\sqrt{2}}=2\sqrt{2};\sqrt[3]{27+27\sqrt{2}+18+2\sqrt{2}}-\sqrt[3]{27-27\sqrt{2}+18-2\sqrt{2}}=3+\sqrt{2}-3+\sqrt{2}=2\sqrt{2};2\sqrt{2}=2\sqrt{2}\)

Ответ: \(2\sqrt{2}=2\sqrt{2}\)

Решение №17099: \(\left ( \frac{\sqrt{3}+1}{1+\sqrt{3}+\sqrt{t}}+\frac{\sqrt{3}-1}{1-\sqrt{3}+\sqrt{t}} \right )\cdot \left ( \sqrt{t}-\frac{2}{\sqrt{t}}+2 \right )=\frac{\left ( \sqrt{3}+1 \right )\left ( 1-\sqrt{3}+\sqrt{t} \right )+\left ( \sqrt{3}-1 \right )\left ( 1+\sqrt{3}+\sqrt{t} \right )}{\left ( \sqrt{t}+1+\sqrt{3} \right )\left ( \sqrt{t}+1-\sqrt{3} \right )}\cdot \frac{\sqrt{t}^{2}-2+2\sqrt{t}}{\sqrt{t}}=\frac{2\sqrt{3\left ( t+2\sqrt{t}-2 \right )}}{t+2\sqrt{t}-2}=2\sqrt{3}\)

Ответ: \(2\sqrt{3}\)

Решение №17100: \(\frac{7-4\sqrt{3}}{\sqrt[3]{26-15\sqrt{3}}}=2-\sqrt{3};\frac{4-4\sqrt{3}+3}{\sqrt[3]{8-12\sqrt{3}+18-3\sqrt{3}}}=\frac{\left ( 2-\sqrt{3} \right )^{2}}{\sqrt[3]{\left ( 2-\sqrt{3} \right )^{3}}}=2-\sqrt{3};2-\sqrt{3}=2-\sqrt{3}\)

Ответ: \(2-\sqrt{3}=2-\sqrt{3}\)

Решение №17101: \(\sqrt{6m+2\sqrt{9m^{2}-n^{2}}}-\sqrt{6m-2\sqrt{9m^{2}-n^{2}}}=2\sqrt{3m-n};\sqrt{3m+n+2\sqrt{\left ( 3m+n \right )\left ( 3m-n \right )}+3m-n}-\sqrt{3m+n-2\sqrt{\left ( 3m+n \right )\left ( 3m-n \right )}+3m-n};\sqrt{\left ( \sqrt{3m+n}+\sqrt{3m-n} \right )^{2}}-\sqrt{\left ( \sqrt{3m+n}-\sqrt{3m-n} \right )^{2}};\sqrt{3m+n}+\sqrt{3m-n}-\left | \sqrt{3m+n}-\sqrt{3m-n} \right |;2\sqrt{3m-n}=2\sqrt{3m-n}\)

Ответ: \(2\sqrt{3m-n}=2\sqrt{3m-n}\)

Решение №17102: \(\sqrt{10p+2\sqrt{25p^{2}-q^{2}}}-\sqrt{10p-2\sqrt{25p^{2}-q^{2}}}=2\sqrt{5p-q};\sqrt{5p+q+2\sqrt{\left ( 5p+q \right )\left ( 5p-q \right )}+5p-q}-\sqrt{5p+q-2\sqrt{\left ( 5p+q \right )\left ( 5p-q \right )}+5p-q}=\sqrt{5p+q}+\sqrt{5p-q}-\left | \sqrt{5p+q}-\sqrt{5p-q} \right |=2\sqrt{5p-q};2\sqrt{5p-q}=2\sqrt{5p-q}\)

Ответ: \(2\sqrt{5p-q}=2\sqrt{5p-q}\)

Решение №17103: \(\frac{\sqrt[3]{a+2\sqrt{a-1}}}{\left ( \sqrt{a-1}+1 \right )^{-\frac{1}{3}}}+\frac{\sqrt[3]{a-2\sqrt{a-1}}}{\left ( \sqrt{a-1}-1 \right )^{-\frac{1}{3}}}=2\sqrt{a-1}=\frac{\sqrt[3]{a-1+2\sqrt{a-1}+1}}{\frac{1}{\sqrt[3]{\sqrt{a-1}+1}}}+\frac{\sqrt[3]{a-1-2\sqrt{a-1}+1}}{\frac{1}{\sqrt{a-1}-1}}=2\sqrt{a-1};\sqrt{a-1}+1+\sqrt{a-1}-1=2\sqrt{a-1};2\sqrt{a-1}-2\sqrt{a-1}\)

Ответ: \(2\sqrt{a-1}-2\sqrt{a-1}\)

Решение №17104: \(4:\left ( 0.6\sqrt[3]{\frac{1}{3}} \right )=10\sqrt[4]{1.5}:\left ( 0.25\sqrt[4]{216\sqrt[3]{9}} \right ); 4:\left ( \frac{3}{5}\cdot \frac{3^{\frac{2}{3}}}{3} \right )=10\sqrt[4]{\frac{3}{2}}:\left ( \frac{1}{4}\cdot \sqrt[4]{2^{3}\cdot 3^{3}\cdot 3^{\frac{2}{3}}} \right ); 4:\frac{3^{\frac{2}{3}}}{5}=\frac{5*3^{\frac{1}{4}}*2^{\frac{3}{4}}*2^{2}}{2^{\frac{3}{4}}*3^{\frac{11}{12}}}; 20*3^{-\frac{2}{3}}=20*3^{-\frac{2}{3}}\)

Ответ: \(20*3^{-\frac{2}{3}}=20*3^{-\frac{2}{3}}\)

Решение №17105: \(\sqrt{8+2\sqrt{10+2\sqrt{5}}}-\sqrt{8-2\sqrt{10+2\sqrt{5}}}=\sqrt{20-4\sqrt{5}};8+2\sqrt{10+2\sqrt{5}}-2\sqrt{\left ( 8+2\sqrt{10+2\sqrt{5}} \right )\left ( 8-2\sqrt{10+2\sqrt{5}} \right )}+8-2\sqrt{10+2\sqrt{5}}=20-4\sqrt{5};16-2\sqrt{24-8\sqrt{5}}=20-4\sqrt{5};4\sqrt{5}-4=2\sqrt{24-8\sqrt{5}};2\sqrt{5}-2=\sqrt{24-8\sqrt{5}};24-8\sqrt{5}=24-8\sqrt{5}\)

Ответ: \(24-8\sqrt{5}\)

Решение №17106: \(\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}=3;9+\sqrt{80}+9-\sqrt{80}+3\sqrt[3]{\left ( 9+\sqrt{80} \right )\left ( 9-\sqrt{80} \right )}\left ( \sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}} \right )=27;18+3\sqrt[3]{81-80}\cdot 3=27;18+3*1*3=27;27=27\)

Ответ: \(27=27\)

Решение №17107: \(\frac{a^{2}-3}{\sqrt{\left ( \frac{a^{2}+3}{2a} \right )^{2}-3}}=\frac{a^{2}-3}{\sqrt{\frac{a^{4}+6a^{2}+9}{4a^{2}}-3}}=\frac{a^{2}-3}{\sqrt{\frac{a^{4}-6a^{2}+9}{4a^{2}}}}=\frac{a^{2}-3}{\left | \frac{a^{2}-3}{2a} \right |}=\frac{2\left ( a^{2}-3 \right )\left | a \right |}{\left ( a^{2}-3 \right )}=-2a;2a\)

Ответ: \(-2a;2a\)

Решение №17108: \(\frac{a+b}{\left ( \sqrt{a}-\sqrt{b} \right )^{2}}\left ( \frac{3ab-b\sqrt{ab}+a\sqrt{ab}-3b^{2}}{\frac{1}{2}\sqrt{\frac{1}{4}}\left ( \frac{a}{b}+\frac{b}{a} \right )^{2}-1}+\frac{4ab\sqrt{a}+9ab\sqrt{b}-9b^{2}\sqrt{a}}{\frac{3}{2}\sqrt{b}-2\sqrt{a}} \right )=\frac{a+b}{\left ( \sqrt{a}-\sqrt{b} \right )^{2}}\left ( \frac{\left ( a-b \right )\left ( 3b+\sqrt{ab} \right )}{\frac{1}{4ab}\sqrt{\left ( a^{2}-b^{2} \right )^{2}}}-2\sqrt{ab}\left ( \sqrt{a}+3\sqrt{b} \right ) \right )=\frac{a+b}{\left ( \sqrt{a}-\sqrt{b} \right )^{2}}\left ( \frac{4ab\left ( a-b \right )\left ( 3b +\sqrt{ab}\right )}{a^{2}-b^{2}}-2\sqrt{ab}\left ( \sqrt{a}+3\sqrt{b} \right ) \right )=\frac{a+b}{a-2\sqrt{ab}+b}\cdot 2\sqrt{ab}\left ( \frac{2\sqrt{a}\left ( 3b+\sqrt{a} \right )}{a+b}-\left ( \sqrt{a}+3\sqrt{b} \right ) \right )=\frac{-2\sqrt{ab}\left ( a\sqrt{a}-5\sqrt{ab}+a\sqrt{b}+3b\sqrt{b} \right )}{a-2\sqrt{ab}+b}=-2\sqrt{ab}\left ( \sqrt{a}+3\sqrt{b} \right )=-2b\left ( a+3\sqrt{ab} \right )\)

Ответ: \(-2b\left ( a+3\sqrt{ab} \right )\)

Решение №17109: \(\frac{11-6\sqrt{2}}{\sqrt[3]{45-29\sqrt{2}}}=3-\sqrt{2};\frac{9-6\sqrt{2}+2}{\sqrt[3]{27-27\sqrt{2}+18-2\sqrt{2}}}=\frac{\left ( 3-\sqrt{2} \right )^{2}}{\left ( 3-\sqrt{2} \right )}=3-\sqrt{2};3-\sqrt{2}=3-\sqrt{2} \)

Ответ: \(3-\sqrt{2}=3-\sqrt{2}\)

Решение №17110: \(19=x^{3}-y^{3}=\left ( x-y \right )\left ( x^{2}+xy+y^{2} \right )=n*m; m=x^{2}+xy+y^{2};\left ( y+1 \right )^{2}+y\left ( y+1 \right )+y^{2}=19;y^{2}+y-6=0;3^{3}-2^{3}\)

Ответ: \(3^{3}-2^{3}\)

Решение №17111: \(\left ( \frac{x-9}{x+3x^{\frac{1}{2}}+9}:\frac{x^{0.5}+3}{x^{1.5}-27} \right )^{0.5}-x^{0.5}=\left ( \frac{x-9}{\sqrt{x}^{2}+3\sqrt{x}+9}:\frac{\sqrt{x}+3}{\sqrt{x}^{2}-3^{3}} \right )^{0.5}-\sqrt{x}=\sqrt{\left ( \sqrt{x}-3 \right )^{2}}-\sqrt{x}=\left | \sqrt{x}-3 \right |-\sqrt{x}=3-2\sqrt{x};-3\)

Ответ: \(3-2\sqrt{x};-3\)

Решение №17112: \(u-v=a;u^{2}-v^{2}=b;u^{3}-v^{3}=c;=u-v=a;\left ( u-v \right )\left ( u+v \right )=b;\left ( u-v \right )\left ( \left ( u+v \right )^{2}-uv \right )=c;3b^{2}+a^{4}=4ac\)

Ответ: \(3b^{2}+a^{4}=4a\)

Решение №17113: \(\left ( \sqrt[4]{mn^{2}p}+m\sqrt{\frac{3n}{m}}+\sqrt{3np} \right )\left ( \sqrt[4]{36mn^{2}p}-\sqrt{3mn}-p\sqrt{\frac{3n}{p}} \right )=\left ( \sqrt[4]{36mn^{2}p}+\left ( \sqrt{3mn}+\sqrt{3np} \right ) \right )\left ( \sqrt[4]{36mn^{2}p}-\left ( \sqrt{3mn}+\sqrt{3np} \right ) \right )=\left ( \sqrt[4]{36mn^{2}p} \right )^{2}-\left ( \sqrt{3mn}+\sqrt{3np} \right )^{2}=\sqrt{36mn^{2}p}-3mn-2\sqrt{9mn^{2}p}-3np=-3n\left ( m+p \right )\)

Ответ: \(-3n\left ( m+p \right )\)

Решение №17114: \(\left ( \frac{1}{\left ( x+3 \right )^{2}}\left ( \frac{1}{x^{2}}+\frac{1}{9} \right )+\frac{2}{\left ( x+3 \right )^{3}} \left ( \frac{1}{x}+\frac{1}{3} \right )\right )^{-\frac{1}{2}}=\left ( \frac{x^{2}+9}{9x^{2}\left ( x+3 \right )^{2}}+\frac{2}{3x\left ( x+3 \right )^{2}} \right )^{-\frac{1}{2}}=\left ( \frac{\left ( x+3 \right )^{2}}{9x^{2}\left ( x+3 \right )^{2}} \right )^{-\frac{1}{2}}=\left ( \frac{1}{9x^{2}} \right )^{-\frac{1}{2}}=\sqrt{9x^{2}}=3\left | x \right |=-3x;-3x\)

Ответ: \(-3x;-3x\)

Решение №17115: \(\left ( x^{2}-6x+1+\left ( \frac{\frac{x-3}{1+3x}-\frac{x-5}{1+5x}}{1+\frac{\left ( x-5 \right )\left ( x-3 \right )}{\left ( 1+5x \right )\left ( 1+3x \right )}} \right )^{-1} \right )^{\frac{1}{2}}=\left ( x^{2}-6x+1+\left ( \frac{\left ( x-3 \right )\left ( 1+5x \right )-\left ( x-5 \right )\left ( 1+3x \right )}{\left ( 1+5x \right )\left ( 1+3x \right )+\left ( x-5 \right )\left ( x-3 \right )} \right )^{-1} \right )^{\frac{1}{2}}=\left ( x^{2}-6x+1+\left ( \frac{2x^{2}+2}{16x^{2}+16} \right )^{-1} \right )^{\frac{1}{2}}=\left ( x^{2}-6x+1+\left ( \frac{1}{8} \right )^{-1} \right )^{\frac{1}{2}}=\left ( x^{2}-6x+1+8 \right )^{\frac{1}{2}}=\sqrt{\left ( x-3 \right )^{2}}=\left | x-3 \right |=3-x;x-3\)

Ответ: \(3-x;x-3\)

Решение №17116: \(\frac{1+\sqrt{1+x}}{x+1}+\frac{1+\sqrt{1-x}}{x-1}=\frac{x+x\sqrt{1+x}-1-\sqrt{x+1}+x+x\sqrt{1-x}+1+\sqrt{1-x}}{x^{2}-1}=\frac{2x-\left ( 1-x \right )\sqrt{1+x}+\left ( 1+x \right )\sqrt{1-x}}{x^{2}-1}=\frac{\sqrt{3}-\left ( 1-\frac{\sqrt{3}}{2} \right )\sqrt{1-\frac{\sqrt{3}}{2}}+\left ( 1+\frac{\sqrt{3}}{2}\sqrt{1-\frac{\sqrt{3}}{2}} \right )}{\frac{3}{4}-1}=-4\sqrt{3}+2-\sqrt{3}+2\sqrt{3}-3+2+\sqrt{3}-2\sqrt{3}-3=-4\sqrt{3}-2\)

Ответ: \(-4\sqrt{3}-2\)

Решение №17117: \(\frac{\sqrt{\left ( 2p+1 \right )^{3}}+\sqrt{\left ( 2p-1 \right )^{3}}}{\sqrt{4p+2\sqrt{4p^{2}-1}}}=\frac{\left ( \sqrt{2p+1}+\sqrt{2p-1} \right )\left ( \left ( \sqrt{2p+1} \right )^{2}-\sqrt{2p+1}\sqrt{2p-1}+\left ( \sqrt{2p-1} \right )^{2}\right ) }{\sqrt{2p+1+2\sqrt{4p^{2}-1}}+2p-1}=\frac{\left ( \sqrt{2p+1}+\sqrt{2p-1} \right )\left ( 4p-\sqrt{4p^{2}-1} \right )}{\sqrt{\left ( \sqrt{2p+1}+\sqrt{2p-1} \right )^{2}}}=4p-\sqrt{4p^{2}-1}\)

Ответ: \(4p-\sqrt{4p^{2}-1}\)

Решение №17118: \(\frac{2\sqrt[3]{2}}{1+\sqrt{3}}=\frac{\sqrt[3]{20+12\sqrt{3}}}{2+\sqrt{3}};\left (\frac{2\sqrt[3]{2}}{1+\sqrt{3}} \right )^{3}=\left (\frac{\sqrt[3]{20+12\sqrt{3}}}{2+\sqrt{3}} \right )^{3};\frac{16}{10+6\sqrt{3}}=\frac{20+12\sqrt{3}}{26+15\sqrt{3}};\frac{2}{5+3\sqrt{3}}=\frac{5+3\sqrt{3}}{26+15\sqrt{3}};2\left ( 26+15\sqrt{3} \right )=\left ( 5+3\sqrt{3} \right )\left ( 5+3\sqrt{3} \right );52+30\sqrt{3}=25+30\sqrt{3}+27;52+30\sqrt{3}=52+30\sqrt{3}\)

Ответ: \(52+30\sqrt{3}=52+30\sqrt{3}\)

Решение №17119: \(\left ( \frac{4}{3-\sqrt{5}} \right )^{2}-\left ( \frac{6-5\sqrt{6}}{5-\sqrt{6}} \right )^{2}=2\sqrt{61+24\sqrt{5}}; \frac{16}{14-6\sqrt{5}}-6=2\sqrt{61+24\sqrt{5}};\frac{8}{7-3\sqrt{5}}-6=2\sqrt{61+24\sqrt{5}}; \frac{4-21+9\sqrt{5}}{7-3\sqrt{5}}=\sqrt{61+24\sqrt{5}};\frac{12\sqrt{5}+16}{4}=\sqrt{61+24\sqrt{5}}; 3\sqrt{5}+4=\sqrt{61+24\sqrt{5}}; 45+24\sqrt{5}+16=61+24\sqrt{5};61+24\sqrt{5}=61+24\sqrt{5}\)

Ответ: \(61+24\sqrt{5}=61+24\sqrt{5}\)

Решение №17120: \(\frac{\sqrt{2}-1}{\sqrt{2}+1}=\sqrt[3]{\frac{10-7\sqrt{2}}{10+7\sqrt{2}}}; \frac{\left ( \sqrt{2}-1 \right )\left ( \sqrt{2}-1 \right )}{\left ( \sqrt{2}+1 \right )\left ( \sqrt{2}-1 \right )}=\sqrt[3]{\frac{\left ( 10-7\sqrt{2} \right )\left ( 10-7\sqrt{2} \right )}{\left ( 10+7\sqrt{2} \right )\left ( 10-7\sqrt{2} \right )}}; 2-2\sqrt{2}+1=\sqrt[3]{\frac{100-140\sqrt{2}+98}{2}};3-2\sqrt{2}=\sqrt[3]{99-70\sqrt{2}};27-54\sqrt{2}+72-12\sqrt{2}=99-70\sqrt{2};99-70\sqrt{2}=99-70\sqrt{2}\)

Ответ: \(99-70\sqrt{2}=99-70\sqrt{2}\)

Решение №17121: \(\frac{a^{\frac{7}{3}}-2a^{\frac{5}{3}}b^{\frac{2}{3}}+ab^{\frac{4}{3}}}{a^{\frac{5}{3}}-a^{\frac{4}{3}}b^{\frac{1}{3}}-ab^{\frac{2}{3}}+a^{\frac{2}{3}}b}:a^{\frac{1}{3}}=\frac{a^{\frac{3}{3}}\left (a^{\frac{4}{3}}-2a^{\frac{2}{3}}b^{\frac{2}{3}}+b^{\frac{4}{3}} \right )}{a^{\frac{2}{3}}\left ( a^{\frac{3}{3}}-a^{\frac{2}{3}}b^{\frac{1}{3}}-a^{\frac{1}{3}}b^{\frac{2}{3}}+b^{\frac{3}{3}} \right )}\cdot \frac{1}{a^{\frac{1}{3}}}=\frac{\left ( a^{\frac{2}{3}}-b^{\frac{2}{3}} \right )^{2}}{\left ( a^{\frac{3}{3}}-a^{\frac{2}{3}}b^{\frac{1}{3}} \right )-\left ( a^{\frac{1}{3}}b^{\frac{2}{3}}+b^{\frac{3}{3}} \right )}=\frac{\left (a^{\frac{2}{3}}-b^{\frac{2}{3}} \right )}{\left ( a^{\frac{1}{3}}-b^{\frac{1}{3}} \right )\left ( a^{\frac{2}{3}}-b^{\frac{2}{3}} \right )}=\frac{a^{\frac{2}{3}}-b^{\frac{2}{3}}}{a^{\frac{1}{3}}-b^{\frac{1}{3}}}=a^{\frac{1}{3}}+b^{\frac{1}{3}}\)

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

Решение №17122: \(\frac{a^{\frac{1}{2}}+ab^{-1}}{a^{-\frac{1}{3}}-a^{-\frac{1}{6}}b^{-\frac{2}{3}}+b^{-\frac{2}{3}}}-\frac{a}{\sqrt[3]{b}}=\frac{\sqrt[6]{a^{5}}\left ( \sqrt[6]{a}+\sqrt[6]{b^{2}}-\sqrt[6]{a} \right )}{\sqrt[6]{b^{2}}}=\frac{\sqrt[6]{a^{5}}\sqrt[6]{b^{2}}}{\sqrt[6]{b^{2}}}=\sqrt[6]{a^{5}}=a^{\frac{5}{6}}\)

Ответ: \(a^{\frac{5}{6}}\)

Решение №17123: \(\left ( \left ( a^{\frac{1}{2}}-b^{\frac{1}{2}} \right )^{-1}\left ( a^{\frac{3}{2}}-b^{\frac{3}{2}} \right )-\frac{1}{\left ( \sqrt{a}+\sqrt{b} \right )^{-2}} \right ):\sqrt[3]{ab\sqrt{ab}}+\frac{1}{1+\left ( a\left ( 1-a^{2} \right )^{-\frac{1}{2}} \right )^{2}}=\left ( \frac{a^{\frac{3}{2}}-b^{\frac{3}{2}}}{a^{\frac{1}{2}}-b^{\frac{1}{2}}}-\left ( \sqrt{a}+\sqrt{b} \right )^{2} \right ):\sqrt{ab}+\frac{1}{1+\frac{a^{2}}{1-a^{2}}}=\left ( a+a^{\frac{1}{2}}b^{\frac{1}{2}}+b-a-2a^{\frac{1}{2}}b^{\frac{1}{2}}-b \right )\cdot \frac{1}{a^{\frac{1}{2}}b^{\frac{1}{2}}}+1-a^{2}=-a^{\frac{1}{2}}b^{\frac{1}{2}}\cdot \frac{1}{a^{\frac{1}{2}}b^{\frac{1}{2}}}+1-a^{2}=1+1-a^{2}=-a^{2}\)

Ответ: \(-a^{2}\)

Решение №17124: \(\frac{\left ( ab\left ( x^{2}+y^{2} \right )+xy\left ( a^{2}+b^{2} \right ) \right )\left ( \left ( ax+by \right )^{2}-4abxy \right )}{ab\left ( x^{2}-y^{2} \right )+xy\left ( a^{2}-b^{2} \right )}=\frac{-\left ( a^{2}-b^{2} \right )y\frac{+}{}y\sqrt{a^{4}-2a^{2}b^{2}+b^{4}+4a^{2}b^{2}}}{2ab}=\frac{\left ( -a^{2}+b^{2}\frac{+}{}\left ( a^{2}+b^{2} \right ) \right )y}{2ab}=\frac{2b^{2}y}{2ab};-\frac{2a^{2}y}{2ab}=-\frac{ay}{b};\frac{by}{a}=abx^{2}+\left ( a^{2}+b^{2} \right )yx+aby^{2}=ab\left ( x+\frac{ay}{b} \right )\left ( x-\frac{by}{a} \right )=\left ( abx+a^{2}y \right )\left ( abx-b^{2}y \right )=a^{2}x^{2}-b^{2}y^{2}\)

Ответ: \(a^{2}x^{2}-b^{2}y^{2}\)

Решение №17125: \(\frac{2a\sqrt{1+x^{2}}}{x+\sqrt{1+x^{2}}}; x=\frac{1}{2}\left ( \sqrt{\frac{a}{b}-\sqrt{\frac{b}{a}}} \right );=\frac{2a\sqrt{1+\left (\frac{1}{2}\left ( \sqrt{\frac{a}{b}-\sqrt{\frac{b}{a}}} \right )^{2}}}{\frac{1}{2}\left ( \sqrt{\frac{a}{b}-\sqrt{\frac{b}{a}}}+\sqrt{1+\left ( \frac{1}{2}\left ( \sqrt{\frac{a}{b}+\sqrt{\frac{b}{a}}} \right )^{2}}}=\frac{2a\sqrt{1+\frac{a^{2}-2ab+b^{2}}{4ab}}}{\frac{a-b}{2\sqrt{ab}}+\sqrt{1+\frac{a^{2}+2ab+b^{2}}{4ab}}}=\frac{a\left ( a+b \right )}{\sqrt{ab}}:\left ( \frac{a-b+a+b}{2\sqrt{ab}} \right )=\frac{a\left ( a+b \right )}{\sqrt{ab}}:\frac{2a}{2\sqrt{ab}}=\frac{a\left ( a+b \right )}{\sqrt{ab}}\cdot \frac{\sqrt{ab}}{a}=a+b\)

Ответ: \(a+b\)

Решение №17126: \(16x^{2}+144x+\left ( a+b \right );b-a=-7;144^{2}-4\cdot 16\left ( a+b \right )=0;b-a=-7;b+a=324;a=165.5;b=158.5\)

Ответ: \(a=165.5;b=158.5\)

Решение №17127: \(\frac{\left ( a^{2}-b^{2} \right )\left ( \sqrt[3]{a}-\sqrt[3]{b} \right )}{\sqrt[3]{a^{4}}+\sqrt[3]{ab^{3}}-\sqrt[3]{a^{3}b}-\sqrt[3]{b^{4}}}=\frac{\left ( a+b \right )\left ( a-b \right )\left ( \sqrt[3]{a}-\sqrt[3]{b} \right )}{\left ( a+b \right )\left ( \sqrt[3]{a}-\sqrt[3]{b} \right )}=a-b\)

Ответ: \(a-b\)

Решение №17128: \(\frac{2b\sqrt{x^{2}-1}}{x-\sqrt{x^{2}-1}}; x=\frac{1}{2}\left ( \sqrt{\frac{a}{b}+\sqrt{\frac{b}{a}}} \right );=\frac{2b\sqrt{\left (\frac{1}{2}\left ( \sqrt{\frac{a}{b}+\sqrt{\frac{b}{a}}} \right )^{2}-1}}{\frac{1}{2}\left ( \sqrt{\frac{a}{b}+\sqrt{\frac{b}{a}}}-\sqrt{\left ( \frac{1}{2}\left ( \sqrt{\frac{a}{b}+\sqrt{\frac{b}{a}}} \right )^{2}-1}}=\frac{2b\sqrt{\frac{\left ( a-b \right )^{2}}{4ab}}}{\frac{a+b}{2\sqrt{ab}-\sqrt{\frac{\left ( a-b \right )^{2}}{4ab}}}}=\frac{b\left ( a-b \right )}{\sqrt{ab}}\cdot \frac{2\sqrt{ab}}{2b}=a-b\)

Ответ: \(a-b\)

Решение №17129: \(\frac{a\left ( a-2 \right )-b\left ( b+2 \right )+\sqrt{ab}\left ( b-a+2 \right )}{a+b-\sqrt{ab}}:\left ( 1+2\frac{a^{2}+b^{2}+ab}{b^{3}-a^{3}} \right )=\frac{a^{2}-2a-b^{2}-2b-\sqrt{ab}\left ( a-b-2 \right )}{a+b-\sqrt{ab}}:\left ( 1-2\frac{a^{2}+b^{2}+ab}{b^{3}-a^{3}} \right )=\frac{\left ( a-b \right )\left ( a+b \right )-2\left ( a+b \right )-\sqrt{ab}\left ( a-b-2 \right )}{a+b-\sqrt{ab}}:\left ( 1-\frac{2}{a-b} \right )=\frac{\left ( a-b-2 \right )\left ( a+b-\sqrt{ab} \right )}{a+b-\sqrt{ab}}\cdot \frac{a-b}{a-b-2}=a-b\)

Ответ: \(a-b\)

Решение №17130: \(\left ( \sqrt[3]{m^{2}}+n\sqrt[3]{m}+n^{2} \right )\cdot \frac{\sqrt[3]{m^{4}}-n^{3}+n^{2}\sqrt[3]{m}-mn}{mn^{-1}+n-n^{4}m^{-1}-n^{2}}=\left ( m^{\frac{2}{3}}+nm^{\frac{1}{3}}+n^{2} \right )\frac{m^{\frac{4}{3}}-n^{3}+n^{2}m^{\frac{1}{3}}-mn}{\frac{m}{n}+n-\frac{n^{4}}{m}-n^{2}}=\left ( m^{\frac{2}{3}}+nm^{\frac{1}{3}}+n^{2} \right )\frac{\left ( m+n^{2} \right )\left ( m^{\frac{1}{3}}-n \right )}{\frac{\left ( m+n^{2} \right )\left ( m-n^{3} \right )}{mn}}=\frac{\left ( m-n^{3} \right )mn}{m-n^{3}}=mn\)

Ответ: \(mn\)

Решение №17131: \(\left ( \frac{pq^{2}}{\left ( p+q \right )^{\frac{5}{2}}}-\frac{2pq^{2}}{\left ( p+q \right )^{\frac{3}{2}}}+\frac{pq}{\sqrt{p+q}} \right ):\left ( \frac{p^{2}}{\left ( p+q \right )^{\frac{5}{2}}}-\frac{p^{2}q}{\left ( p+q \right )^{\frac{7}{2}}} \right )=\frac{pq}{\left ( p+q \right )^{\frac{1}{2}}}\cdot \left ( \frac{q^{2}}{\left ( p+q \right )^{2}} -\frac{2q}{p+q}+1\right ):\frac{p^{2}}{\left ( p+q \right )^{\frac{5}{2}}}\cdot \left ( 1-\frac{q}{p+q} \right )=\frac{pq}{\left ( p+q \right )^{\frac{1}{2}}}\left ( \frac{q^{2}-2q\left ( p+q \right )+\left ( p+q \right )^{2}}{\left ( p+q \right )^{2}} \right ):\frac{p^{2}}{\left ( p+q \right )^{\frac{5}{2}}}\left ( \frac{p+q-q}{p+q} \right )=\frac{p^{3}q}{\left ( p+q \right )^{\frac{5}{2}}}\cdot \frac{\left ( p+q \right )^{\frac{5}{2}}\left ( p+q \right )}{p^{3}}=q\left ( p+q \right )\)

Ответ: \(q\left ( p+q \right )\)

Решение №17132: \(\frac{\sqrt{1+\sqrt{1-x^{2}}}\left ( \sqrt{\left ( 1+x \right )^{3}}-\sqrt{\left ( 1-x \right )^{3}} \right )}{2+\sqrt{1-x^{2}}}=\sqrt{1+\sqrt{1-x^{2}}}\left ( \sqrt{1+x}-\sqrt{1-x} \right )=\sqrt{\frac{1+x+2\sqrt{1-x^{2}+1-x}}{2}}\left ( \sqrt{1+x}-\sqrt{1-x} \right )=\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{2}}\left ( \sqrt{1+x}-\sqrt{1-x} \right )=\frac{1+x-1+x}{\sqrt{2}}=\frac{2x}{\sqrt{2}}=x\sqrt{2}\)

Ответ: \(x\sqrt{2}\)

Решение №17133: \(x\sqrt[3]{2x\sqrt{xy}-x\sqrt{3xy}}\cdot \sqrt[6]{x^{2}y\left ( 7+4\sqrt{3} \right )}=x\sqrt[3]{x\sqrt{xy\left ( 2-\sqrt{3} \right )}}\cdot \sqrt[6]{x^{3}y\left ( 7+4\sqrt{3} \right )}=x\sqrt[3]{x\sqrt{xy}\left ( 2-\sqrt{3} \right )}\cdot \sqrt[6]{\left | x \right |\sqrt{xy}\left ( 2+\sqrt{3} \right )}=x\sqrt[3]{x\left | x \right |\left | x \right |\left | y \right |}=x\sqrt[3]{x^{3}\left | y \right |}=x\cdot x\cdot \left | \sqrt[3]{y} \right |=x^{2}\left | \sqrt[3]{y} \right |\)

Ответ: \(x^{2}\left | \sqrt[3]{y} \right |\)

Решение №17134: \(\frac{x^{4}+x^{2}+x\sqrt{2}+2}{x^{2}-x\sqrt{2}+2}-x\sqrt{2}=\frac{x^{4}+x^{2}+x\sqrt{2}+2-x^{3}\sqrt{2}+2x^{2}-2x\sqrt{2}}{x^{2}-x\sqrt{2}+2}=\frac{x^{4}-\sqrt{2}x^{3}+3x^{2}-\sqrt{2}x+2}{x^{2}-x\sqrt{2}+2}=\frac{\left ( x^{2}-\sqrt{2}x+2 \right )\left ( x^{2}+1 \right )}{x^{2}-x\sqrt{2}+2}=x^{2}+1\)

Ответ: \(x^{2}+1\)

Решение №17135: \(\frac{\left ( \sqrt{a}+\sqrt{ax}+x+x\sqrt{x} \right )^{2}\left ( 1-\sqrt{x} \right )^{2}}{\left ( x+x^{-1}-2 \right )a^{-\frac{1}{4}}}-\frac{\left ( x\sqrt{a} \right )^{\frac{3}{2}}}{\left ( ax^{-1}+4\sqrt{a}+4x \right )^{-\frac{1}{2}}}=\frac{\left ( \sqrt{a}\left ( 1+\sqrt{x} \right ) +x\left ( 1+\sqrt{x} \right )\right )^{2}\left ( 1-\sqrt{x} \right )^{2}}{\left ( x+\frac{1}{x}-2 \right )\frac{1}{\sqrt[4]{a}}}-\frac{\sqrt{x^{3}}\sqrt[4]{a^{3}}}{\left ( \frac{a}{x}+4\sqrt{a}+4x \right )^{-\frac{1}{2}}}=\left ( \sqrt{a}+x \right )^{2}\cdot x\sqrt[4]{a}-x\sqrt[4]{a^{3}}\left ( \sqrt{a}+2x \right )=x\sqrt[4]{a}\left ( \left ( \sqrt{a}+x \right )^{2}-\sqrt[4]{a^{2}}\left ( \sqrt{a}+2x \right ) \right )=x\sqrt[4]{a}\left ( a+2\sqrt{a}\cdot x+x^{2}-a-2\sqrt{a}\cdot x \right )=x\sqrt[4]{a}\cdot x^{2}=x^{3}\cdot \sqrt[4]{a}\)

Ответ: \(x^{3}\cdot \sqrt[4]{a}\)

Решение №17136: \(\frac{x-1}{x+x^{\frac{1}{2}}+1}:\frac{x^{0.5}+1}{x^{1.5}-1}+\frac{2}{x^{-0.5}}=\frac{\left ( x^{\frac{1}{2}}-1 \right )\left ( x^{\frac{1}{2}}+1 \right )}{x+x^{\frac{1}{2}}+1}\cdot \frac{\left ( x^{\frac{1}{2}} \right )^{3}-1}{x^{\frac{1}{2}}+1}+\frac{2}{\frac{1}{x^{\frac{1}{2}}}}=\frac{\left ( x^{\frac{1}{2}}-1 \right )\left ( x^{\frac{1}{2}}+1 \right )\left (x+x^{\frac{1}{2}}+1 \right )}{x+x^{\frac{1}{2}}+1}+2x^{\frac{1}{2}}=\left ( x^{\frac{1}{2}}-1 \right )^{2}+2x^{\frac{1}{2}}=x-2x\frac{1}{2}+1+2x\frac{1}{2}=x+1\)

Ответ: \(x+1\)

Решение №17137: \(\frac{\sqrt[4]{x^{5}}+\sqrt[4]{xy^{4}}-\sqrt[4]{x^{4}y}-\sqrt[4]{x^{5}}}{\sqrt{x}-\sqrt{y}}\cdot \left ( \sqrt[4]{x}+\sqrt[4]{y} \right )=\frac{\sqrt[4]{x}\left ( \sqrt[4]{x^{4}}+\sqrt[4]{y^{4}} \right )-\sqrt[4]{y}\left ( \sqrt[4]{x^{4}}+\sqrt[4]{y^{4}} \right )}{\left ( \sqrt[4]{x}-\sqrt[4]{y} \right )\left ( \sqrt[4]{x}+\sqrt[4]{y} \right )}\cdot \left ( \sqrt[4]{x}+\sqrt[4]{y} \right )=\sqrt[4]{x^{4}}+\sqrt[4]{y^{4}}=x+y\)

Ответ: \(x+y\)

Решение №17138: \(\left ( \left ( \frac{2^{\frac{3}{2}}+27y^{\frac{3}{5}}}{\sqrt{2}+3\sqrt[5]{y}}+3\sqrt[10]{32y^{2}}-2 \right )\cdot 3^{-2} \right )^{5}=\left ( \left ( \frac{\left ( \sqrt{2} \right )^{3}+\left ( 3\sqrt[5]{y} \right )^{3}}{\sqrt{2}+3\sqrt[5]{y}}+3\sqrt{2}\sqrt[5]{y}-2 \right )\cdot \frac{1}{9} \right )^{5}=\left ( \left ( \frac{\left ( \sqrt{2}+3\sqrt[5]{y} \right )\left ( \left ( \sqrt{2} \right )^{2}-3\sqrt{2}\sqrt[5]{y}+\left ( 3\sqrt[5]{y} \right )^{2} \right )}{\sqrt{2}+3\sqrt[5]{y}}+3\sqrt{2}\sqrt[5]{y}-2 \right ) \cdot \frac{1}{9}\right )^{5}=\left ( 9\sqrt[5]{y} \cdot \frac{1}{9}\right )^{5}=\left ( \sqrt[5]{y} \right )^{5}=y^{2}\)

Ответ: \(y^{2}\)