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