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 №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 №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: \(\frac{-1}{a^{2}+a+1}\)

Решение №16957: \(\frac{2\left | a+5 \right |-a+\frac{25}{a}}{3a^{2}+10a-25}=\frac{2a\left | a+5 \right |-a^{2}+25}{a\left ( 3a^{2}+10a-25 \right )}=\frac{3\left ( 3a^{2}+10a-25 \right )}{a\left ( 3a^{2}+10a-25 \right )};\frac{a^{2}+10a+25}{a\left ( a+5 \right )\left ( 3a-5 \right )}=-\frac{1}{a};\frac{a+5}{a\left ( 3a-5 \right )}\)

Ответ: \(-\frac{1}{a};\frac{a+5}{a\left ( 3a-5 \right )}\)

Решение №16958: \(\left ( \frac{\sqrt[4]{a^{3}}-1}{\sqrt[4]{a}-1}+\sqrt[4]{a} \right )^{\frac{1}{2}}\cdot \left ( \frac{\sqrt[4]{a^{3}}+1}{\sqrt[4]{a}+1}-\sqrt{a} \right )\cdot \left ( a-\sqrt{a^{3}} \right )^{-1}=\left ( \frac{\left ( \sqrt[4]{a}-1 \right )\left ( \sqrt[4]{a^{2}}+\sqrt[4]{a}+1 \right )}{\sqrt[4]{a}-1}+\sqrt[4]{a} \right )^{\frac{1}{2}}\cdot \left ( \frac{\left ( \sqrt[4]{a}+1 \right )\left ( \sqrt[4]{a^{2}}-\sqrt[4]{a}+1 \right )}{\sqrt[4]{a}+1}-\sqrt{a} \right )\cdot \frac{1}{a-\sqrt{a^{3}}}=\left ( \sqrt[4]{a^{2}}+\sqrt[4]{a}+1+\sqrt[4]{a} \right )^{\frac{1}{2}}\cdot \left ( \sqrt[4]{a^{2}}-\sqrt[4]{a}+1-\sqrt[4]{a^{2}} \right )\cdot \frac{1}{a-\sqrt{a^{3}}}=\frac{\left ( \sqrt[4]{a}+1 \right )\left ( 1-\sqrt[4]{a} \right )}{a\left ( 1-\sqrt[4]{a} \right )\left ( 1+\sqrt[4]{a} \right )}=\frac{1}{a}\)

Ответ: \(\frac{1}{a}\)

Решение №16959: \(\frac{a^{2}-4-\left | a-2 \right |}{a^{3}+2a^{2}-5a-6}=\frac{\left ( a-2 \right )\left ( a+2 \right )-\left | a-2 \right |}{\left ( a-2 \right )\left ( a+3 \right )\left ( a+1 \right )}=\frac{a+3}{\left ( a+3 \right )\left ( a+1 \right )};\frac{a+1}{\left ( a+3 \right )\left ( a+1 \right )}=\frac{1}{a+1};\frac{1}{a+3}\)

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

Решение №16960: \(\frac{\left ( \sqrt{a}+\sqrt{b} \right )^{2}-4b}{\left ( a-b \right ):\left ( \sqrt{\frac{1}{b}}+3\sqrt{\frac{1}{a}} \right )}:\frac{a+9b+6\sqrt{ab}}{\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{a}}}=\frac{\frac{a+2\sqrt{ab}+b-4b}{\left ( \sqrt{a}-\sqrt{b} \right )\left ( \sqrt{a}+\sqrt{b} \right ):\left ( \frac{1}{\sqrt{b}}+\frac{3}{\sqrt{a}} \right )}}{\frac{\left ( \sqrt{a}+3\sqrt{b} \right )^{2}}{\frac{\sqrt{a}+\sqrt{b}}{\sqrt{ab}}}}=\frac{a+2\sqrt{ab}-3b}{\left ( \sqrt{a}-\sqrt{b} \right )\left ( \sqrt{a}+\sqrt{b} \right ):\frac{\sqrt{a}+3\sqrt{b}}{\sqrt{ab}}}:\frac{\left ( \sqrt{a}+3\sqrt{b} \right )^{2}\sqrt{ab}}{\left ( \sqrt{a}+\sqrt{b} \right )}=\frac{a+2\sqrt{ab}-3b}{ab\left ( a-\sqrt{ab}+3\sqrt{ab-3b} \right )}=\frac{1}{ab}\)

Ответ: \(\frac{1}{ab}\)

Решение №16961: \(\left ( \left ( \frac{a\sqrt[3]{b}}{b\sqrt{a^{3}}} \right )^{\frac{3}{2}}+\left ( \frac{\sqrt{a}}{a\sqrt[8]{b^{3}}} \right )^{2} \right ):\left ( a^{\frac{1}{4}}+b^{\frac{1}{4}} \right )=\left ( \frac{a^{\frac{3}{2}}\left ( \sqrt[3]{b} \right )^{\frac{3}{2}}}{b^{\frac{3}{2}}\left ( \sqrt{a^{3}} \right )^{\frac{3}{2}}}+\frac{\left ( \sqrt{a} \right )^{2}}{a^{2}\left ( \sqrt[8]{b^{3}} \right )^{2}} \right ):\left ( a^{\frac{1}{4}}+b^{\frac{1}{4}} \right )=\left ( \frac{a^{\frac{3}{2}}b^{\frac{1}{2}}}{a^{\frac{9}{4}}b^{\frac{3}{2}}}+\frac{a}{a^{2}b^{\frac{3}{4}}} \right )\frac{1}{a^{\frac{1}{4}}+b^{\frac{1}{4}}}=\left ( \frac{1}{a^{\frac{3}{4}}b}+\frac{1}{ab^{\frac{3}{4}}} \right )\frac{1}{a^{\frac{1}{4}}+b^{\frac{1}{4}}}=\frac{1}{ab}\)

Ответ: \(\frac{1}{ab}\)

Решение №16963: \(\frac{m\left | m-3 \right |}{\left ( m^{2}-m-6 \right )\left | m \right |}; 1)\frac{m\left | m-3 \right |}{\left ( m^{2}-m-6 \right )\left | m \right |}=\frac{-m\left ( m-3 \right )}{-\left ( m^{2}-m-6 \right )m}=\frac{m-3}{\left ( m-3 \right )\left ( m+2 \right )}=\frac{1}{m+2} 2)\frac{m\left | m-3 \right |}{\left ( m^{2}-m-6 \right )\left | m \right |}=\frac{-m\left ( m-3 \right )}{\left ( m^{2}-m-6 \right )m}=\frac{-m+3}{\left ( m-3 \right )\left ( m+2 \right )}=-\frac{1}{m+2} 3)\frac{m\left | m-3 \right |}{\left ( m^{2}-m-6 \right )\left | m \right |}=\frac{-m\left ( m-3 \right )}{-\left ( m^{2}-m-6 \right )m}=\frac{1}{m+2} \frac{1}{m+2};-\frac{1}{m+2}\)

Ответ: \(\frac{1}{m+2};-\frac{1}{m+2}\)

Решение №16964: \(\frac{\sqrt{4x+4+x^{-1}}}{\sqrt{x}\left | 2x^{2}-x-1 \right |}=\frac{\sqrt{4x+4+\frac{1}{x}}}{\sqrt{x}\left | \left ( x-1 \right )\left ( 2x+1 \right ) \right |}=\frac{\sqrt{\left ( 2x+1 \right )^{2}}}{x\left | \left ( x-1 \right )\left ( 2x+1 \right ) \right |}=\frac{2x+1}{x\left ( 2x+1 \right )\left | x-1 \right |}=\frac{1}{x\left | x-1 \right |}=-\frac{1}{x\left ( x-1 \right )};\frac{1}{x\left ( x-1 \right )}\)

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

Решение №16965: \(\frac{\left ( \sqrt{x}+\sqrt{2} \right )^{2}-\sqrt{2x}}{x^{2}+x-\sqrt{2x}+2}=\frac{x^{2}+2\sqrt{2x}+2-\sqrt{2x}}{\left ( x^{2}+x\sqrt{2x}+2x \right )-\left ( x\sqrt{2x}+\sqrt{2x}\sqrt{2x}+2\sqrt{2x} \right )+\left ( x+\sqrt{2x}+2 \right )}=\frac{x+\sqrt{2x}+2}{x\left ( x+\sqrt{2x}+2 \right )-\sqrt{2x}\left ( x+\sqrt{2x}+2 \right )+\left ( x+\sqrt{2x}+2 \right )}=\frac{x+\sqrt{2x}+2}{\left ( x+\sqrt{2x}+2 \right )\left ( x-\sqrt{2x}+1 \right )}=\frac{1}{x-\sqrt{2x}+1}\)

Ответ: \(\frac{1}{x-\sqrt{2x}+1}\)

Решение №16966: \(\left ( x\sqrt[3]{\frac{x-1}{\left ( x+1 \right )^{2}}}+\frac{x-1}{\sqrt[3]{\left ( x^{2}-1 \right )^{2}}} \right )^{-\frac{3}{5}}:\left ( x^{2}-1 \right )^{\frac{4}{5}}=\left ( \frac{x\sqrt[3]{x-1}}{\sqrt[3]{\left ( x+1 \right )^{2}}}+\frac{\sqrt[3]{x-1}}{\sqrt[3]{\left ( x+1 \right )^{2}}} \right )^{-\frac{3}{5}}\cdot \frac{1}{\sqrt[5]{\left ( x+1 \right )^{4}}}=\left ( \frac{\sqrt[3]{x-1}\left ( x+1 \right )}{\sqrt[3]{\left ( x+1 \right )^{2}}} \right )^{-\frac{3}{5}}\cdot \frac{1}{\sqrt[5]{\left ( x^{2}-1 \right )^{4}}}=\left ( \sqrt[3]{\left ( x-1 \right )\left ( x+1 \right )} \right )^{-\frac{3}{5}}\cdot \frac{1}{\sqrt[5]{\left ( x^{2}-1 \right )^{4}}}=\frac{1}{\sqrt[5]{\left ( x^{2}-1 \right )^{5}}}=\frac{1}{x^{2}-1}\)

Ответ: \(\frac{1}{x^{2}-1}\)

Решение №16967: \(\frac{x^{3}-6x^{2}+11x-6}{\left ( x^{3}-4x^{2}+3x \right )\left | x-2 \right |}=\frac{\left ( x-3 \right )\left ( x-2 \right )\left ( x-1 \right )}{x\left ( x-3 \right )\left ( x-1 \right )\left | x-2 \right |}=\frac{x-2}{x\left | x-2 \right |}=-\frac{1}{x};\frac{1}{x}\)

Ответ: \(-\frac{1}{x};\frac{1}{x}\)

Решение №16968: \(\left ( \frac{z-2}{6z+\left ( z-2 \right )^{2}}+\frac{\left ( z+4 \right )^{2}-12}{z^{3}-8}-\frac{1}{z-2} \right ):\frac{z^{3}+2z^{2}+2z+4}{z^{3}-2z^{2}+2z-4}=\left ( \frac{z-2}{6z+z^{2}-4z+4}+\frac{z^{2}+8z+16-12}{\left ( z-2 \right )\left ( z^{2}+2z+4 \right )}-\frac{2}{z-2} \right ):\frac{\left ( z+2 \right )\left ( z^{2}+2 \right )}{\left ( z-2 \right )\left ( z^{2}+2 \right )}=\frac{z^{2}+2z+4}{\left ( z+2 \right )\left ( z^{2}+2 \right )}\cdot \frac{z-2}{z+2}=\frac{1}{z+2}\)

Ответ: \(\frac{1}{z+2}\)