Решение №17027: \(\left ( \frac{a}{b}\sqrt[3]{b-\frac{4a^{6}}{b^{3}}}-a^{2}\sqrt[3]{\frac{b}{a^{6}}-\frac{4}{b^{3}}}+\frac{2}{ab}\sqrt[3]{a^{3}b^{4}-4a^{9}} \right ):\frac{\sqrt[3]{b^{2}-2a^{3}}}{b^{2}}=\left ( \frac{a}{b}\sqrt[3]{\frac{b^{4}-4a^{6}}{^{3}}}-a^{2}\sqrt[3]{\frac{b^{4}-4a^{6}}{a^{6}b^{3}}}+\frac{2}{ab}\sqrt[3]{a^{3}\left ( b^{4}-4a^{6} \right )} \right )\cdot \frac{b^{2}}{\sqrt[3]{b^{2}-2a^{3}}}=\left ( \frac{a\sqrt[3]{b^{4}-4a^{6}}}{b^{2}} -\frac{a^{2}\sqrt[3]b^{4}-4a^{6}}{{a^{2}b}}+\frac{2a\sqrt[3]{b^{4}-4a^{6}}}{ab}\right )\cdot \frac{b^{2}}{\sqrt[3]{b^{2}-2a^{3}}}=\frac{\sqrt[3]{\left ( b^{2}-2a^{3} \right )\left ( b^{2}+2a^{3} \right )\left ( a+b \right )}}{b^{2}}\cdot \frac{b^{2}}{\sqrt[3]{b^{2}-2a^{3}}}=\left ( a+b \right )\sqrt[3]{b^{2}+2a^{3}}\)

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

Решение №17028: \(4ab+\frac{\left ( 1+\left ( \frac{a}{b} \right )^{-3} \right )a^{3}}{\left ( \sqrt{a}+\sqrt{b} \right )^{2}-2\sqrt{ab}}-\frac{\left ( \frac{\sqrt{a}+\sqrt{b}}{2b\sqrt{a}} \right )^{-1}+\left ( \frac{\sqrt{a}+\sqrt{b}}{2a\sqrt{b}} \right )^{-1}}{\left ( \frac{a+\sqrt{ab}}{2} \right )^{-1}+\left ( \frac{b+\sqrt{ab}}{2} \right )^{-1}}=4ab+\frac{a^{3}+b^{3}}{a+b}-\frac{\frac{2b\sqrt{a}+2a\sqrt{b}}{\sqrt{a}+\sqrt{b}}}{\frac{2b+2\sqrt{ab}+2a+2\sqrt{ab}}{\left ( a+\sqrt{ab} \right )\left ( b+\sqrt{ab} \right )}}=4ab+a^{2}-ab+b^{2}-\sqrt{ab}\frac{\sqrt{ab}\left ( \sqrt{a}+\sqrt{b} \right )}{\left ( \sqrt{a}+\sqrt{b} \right )^{2}}=a^{2}+3ab+b^{2}-ab=a^{2}+2ab+b^{2}=\left ( a+b \right )^{2}\)

Ответ: \(\left ( a+b \right )^{2}\)

Решение №17029: \(\frac{m^{5}+m^{4}\sqrt[3]{2}+\sqrt[3]{4m^{9}}}{\left | m^{3}-1 \right |-1}=\frac{m^{5}+\sqrt[3]{2}m^{4}+\sqrt[3]{2^{2}}m^{3}}{\left | m^{3}-1 \right |-1}=\frac{m^{3}\left ( m^{2}+\sqrt[3]{2}m+\sqrt[3]{2^{2}} \right )}{\left | m^{3}-1 \right |-1}=-\left ( m^{2}+\sqrt[3]{2}m+\sqrt[3]{2^{2}} \right );\frac{m^{3}}{m-\sqrt[3]{2}}\)

Ответ: \(-\left ( m^{2}+\sqrt[3]{2}m+\sqrt[3]{2^{2}} \right );\frac{m^{3}}{m-\sqrt[3]{2}}\)

Решение №17030: \(\frac{x^{4}-x^{3}-x+1}{x^{3}-5x^{2}+7x-3}\left | x-3 \right |=\frac{\left ( x-1 \right )^{2}\left ( x^{2}+x+1 \right )}{\left ( x-1 \right )^{2}\left ( x-3 \right )}\left | x-3 \right |=\frac{\left ( x^{2}+x+1 \right )\left | x-3 \right |}{x-3}=-\left ( x^{2}+x+1 \right ); x^{2}+x+1 \)

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

Решение №17031: \(x^{8}-16=\left ( x^{4}-4 \right )\left ( x^{4}+4 \right )=\left ( x^{2}-2 \right )\left ( x^{2}+2 \right )\left ( x^{4}+4x^{2}+4-4x^{2} \right )=\left ( x^{2}-2 \right )\left ( x^{2}+2 \right )\left ( \left ( x^{2}+2 \right )^{2} -4x^{2}\right )=\left ( x^{2}-2 \right )\left ( x^{2}+2 \right )\left ( \left ( x^{2}+2 \right )^{2}-\left ( 2x \right )^{2} \right )=\left ( x^{2}-2 \right )\left ( x^{2}+2 \right )\left ( x^{2}+2-2x \right )\left ( x^{2}+2+2x \right )=\left ( x^{2}-2 \right )\left ( x^{2}+2 \right )\left ( x^{2}-2x+2 \right )\left ( x^{2}+2x+2 \right )\)

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

Решение №17032: \(\sqrt{\left ( y^{2}+\frac{4}{y^{2}} \right )^{2}-8\left ( y+\frac{2}{y} \right )^{2}+48}=\sqrt{\left ( y+\frac{2}{y} \right )^{4}-8\left ( y+\frac{2}{y} \right )^{2}+16-8\left ( y+\frac{2}{y} \right )^{2}+48}=\sqrt{\left ( y+\frac{2}{y} \right )^{4}-16\left ( y+\frac{2}{y} \right )^{2}+64}=\sqrt{\left ( y^{2}+4+\frac{4}{y^{2}}-8 \right )^{2}}=\sqrt{\left ( \left ( y-\frac{2}{y} \right )^{2} \right )^{2}}=\left ( y-\frac{2}{y} \right )^{2}\)

Ответ: \(\left ( y-\frac{2}{y} \right )^{2}\)

Решение №17033: \(\left ( \frac{\left ( z^{\frac{2}{p}}+z^{\frac{2}{q}} \right )^{2}-4z^{\frac{2}{p}+\frac{2}{q}}}{\left ( z^{\frac{1}{p}}-z^{\frac{1}{q}} \right )^{2}+4z^{\frac{1}{p}+\frac{1}{q}}} \right )^{\frac{1}{2}}=\left ( \frac{\left ( z^{\frac{2}{p}}-z^{\frac{2}{q}} \right )}{\left ( z^{\frac{1}{p}}+z^{\frac{1}{q}} \right )^{2}} \right )^{\frac{1}{2}}=\left | z^{\frac{1}{p}}-z^{\frac{1}{q}} \right |\)

Ответ: \(\left | z^{\frac{1}{p}}-z^{\frac{1}{q}} \right |\)

Решение №17034: \(\sqrt[4]{32\sqrt[3]{4}}+\sqrt[4]{64\sqrt[3]{\frac{1}{2}}}-3\sqrt[3]{2\sqrt[4]{2}}=\sqrt[4]{2^{5}\cdot 2^{\frac{2}{3}}}+\sqrt[4]{2^{6}\cdot 2^{-\frac{1}{3}}}-3\sqrt[3]{2\cdot 2^{\frac{1}{4}}}=2^{\frac{17}{12}}+2^{\frac{17}{12}}-3\cdot 2^{\frac{5}{12}}=2\cdot 2^{\frac{17}{12}}-3\cdot 2^{\frac{5}{12}}=2^{\frac{5}{12}}\left ( 4-3 \right )=2^{\frac{5}{12}}=\sqrt[12]{32}\)

Ответ: \(\sqrt[12]{32}\)

Решение №17035: \(\sqrt[3]{\frac{2x^{2}}{9+18x+9x^{2}}}\sqrt{\frac{\left ( x+1 \right )\sqrt[3]{1-x}}{x}}\sqrt[3]{\frac{3\sqrt{1-x^{2}}}{2x\sqrt{x}}}=\sqrt[6]{\frac{4x^{4}}{81\left ( 1+x \right )^{4}}\cdot \frac{\left ( 1+x \right )^{3}\left ( 1-x \right )}{x^{3}}\frac{9\left ( 1-x^{2} \right )}{4x^{3}}}=\sqrt[6]{\frac{36x^{4}\left ( 1+x \right )^{4}\left ( 1-x \right )^{2}}{324x^{6}\left ( 1+x \right )^{4}}}=\sqrt[6]{\frac{\left ( 1-x \right )^{2}}{9x^{2}}}=\sqrt[3]{\frac{1-x}{3x}}\)

Ответ: \(\sqrt[3]{\frac{1-x}{3x}}\)

Решение №17036: \(\left ( \sqrt{\frac{\left ( 1-n \right )^{3}\sqrt{1+n}}{n}}\sqrt[3]{\frac{3n^{2}}{4-8n+4n^{2}}} \right )^{-1}:\sqrt[3]{\left ( \frac{3n\sqrt{n}}{2\sqrt{1-n^{2}}} \right )^{-1}}=\left ( \sqrt[6]{\left ( \frac{\left ( 1-n \right )^{3}\sqrt[3]{1+n}}{n^{3}} \right )^{3}}\sqrt[6]{\frac{3n^{2}}{4\left ( 1-2n+n^{2} \right )^{2}}} \right )^{-1}\sqrt[6]{\left ( \frac{3n\sqrt{n}}{2\sqrt{1-n^{2}}} \right )^{2}}=\left ( \sqrt[6]{\frac{\left ( 1-n \right )^{3}\left ( 1+n \right )9n^{4}}{n^{3}16\left ( 1-n \right )^{4}}} \right )^{-1}\sqrt[6]{\frac{9n^{3}}{4\left ( 1-n \right )\left ( 1+n \right )}}=\sqrt[6]{\frac{16\left ( 1-n \right )9n^{3}}{9n\left ( 1+n \right )4\left ( 1-n \right )\left ( 1+n \right )}}=\sqrt[6]{\frac{4n^{2}}{\left ( 1+n \right )^{2}}}=\sqrt[3]{\frac{2n}{1+n}}\)

Ответ: \(\sqrt[3]{\frac{2n}{1+n}}\)