Решение №16927: \(\sqrt[4]{\left ( 1-2a+a^{2} \right )\left ( a^{2}-1 \right )\left ( a-1 \right )}:\frac{a^{2}+2a-3}{\sqrt[4]{a+1}}=\sqrt[4]{\left ( a-1 \right )^{2}\left ( a-1 \right )\left ( a+1 \right )\left ( a-1 \right )}\cdot \frac{\sqrt[4]{a+1}}{a^{2}+2a-3}=\frac{\sqrt[4]{\left ( a-1 \right )^{4}\left ( a+1 \right )^{2}}}{\left ( a-1 \right )\left ( a+3 \right )}=\frac{\left | a-1 \right |\sqrt[4]{\left ( a+1 \right )^{2}}}{\left ( a-1 \right )\left ( a+3 \right )}=\frac{\left | a-1 \right |\sqrt{\left ( a+1 \right )}}{\left ( a-1 \right )\left ( a+3 \right )}=-\frac{\sqrt{a+1}}{a+3};\frac{\sqrt{a+1}}{a+3}\)

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

Решение №16928: \(\frac{\sqrt{2a+2\sqrt{a^{2}-b^{2}}}-\sqrt{a-b}}{\sqrt{2a-2\sqrt{a^{2}-b^{2}}}+\sqrt{a-b}}=\frac{\sqrt{a+b}+\sqrt{a-b}-\sqrt{a-b}}{\left | \sqrt{a+b}-\sqrt{a-b} \right |+\sqrt{a-b}}=\frac{\sqrt{a+b}}{\left | \sqrt{a+b}-\sqrt{a-b} \right |+\sqrt{a-b}}=\frac{\sqrt{a+b}}{2\sqrt{a-b}-\sqrt{a+b}};\frac{\sqrt{a+b}}{\sqrt{a+b}}=\frac{\sqrt{a+b}}{2\sqrt{a-b}-\sqrt{a+b}};1\)

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

Решение №16929: \(\frac{\left ( m-1 \right )\sqrt{m}-\left ( n-1 \right )\sqrt{n}}{\sqrt{m^{3}n}+mn+m^{2}-m}=\frac{m\sqrt{m}-\sqrt{m}-n\sqrt{n}+\sqrt{n}}{m\sqrt{mn}+mn+m^{2}-m}=\frac{\left ( \sqrt{m^{3}}-\sqrt{n^{3}} \right )-\left ( \sqrt{m}-\sqrt{n} \right )}{m\left ( \sqrt{mn}+n+m-1 \right )}=\frac{\left ( \sqrt{m}-\sqrt{n} \right )\left ( m+\sqrt{mn}+n-1 \right )}{m\left ( \sqrt{mn}+n+m-1 \right )}=\frac{\sqrt{m}-\sqrt{n}}{m}\)

Ответ: \(\frac{\sqrt{m}-\sqrt{n}}{m}\)

Решение №16930: \(\frac{\sqrt{\frac{p^{4}+q^{4}}{p^{4}-p^{2}q^{2}}+\frac{2q^{2}}{p^{2}-q^{2}}\left ( p^{3}-pq^{2} \right )-2q\sqrt{p}}}{\sqrt{\frac{p}{p-q}}-\frac{q}{p+q}-\frac{2pq}{p^{2}-q^{2}}\cdot \left ( p-q \right )}=\frac{p-q}{\sqrt{p}}\sqrt{\frac{p+q}{p-q}}=\frac{1}{\sqrt{p}}\sqrt{\frac{\left ( p-q \right )^{2}\left ( p+q \right )}{p-q}}=\frac{1}{\sqrt{p}}\sqrt{\left ( p-q \right )\left ( p+q \right )}=\frac{\sqrt{p^{2}-q^{2}}}{\sqrt{p}}\)

Ответ: \(\frac{\sqrt{p^{2}-q^{2}}}{\sqrt{p}}\)

Решение №16931: \(\left ( \frac{t\sqrt{t+2}}{\sqrt{t-2}}-\frac{2\sqrt{t-2}}{\sqrt{t+2}}-\frac{4t}{\sqrt{t^{2}-4}} \right )^{\frac{1}{2}}:\sqrt[4]{t^{2}-4}=\left ( \frac{t\left ( t+2 \right )-2\left ( t-2 \right )-4t}{\sqrt{t^{2}-4}} \right )^{\frac{1}{2}}:\sqrt[4]{t^{2}-4}=\left ( \frac{t^{2}-4t+4}{\sqrt{t^{2}-4}} \right )^{\frac{1}{2}}\cdot \frac{1}{\sqrt[4]{t^{2}-4}}=\frac{t-2}{\sqrt[4]{t^{2}-4}}\cdot \frac{1}{\sqrt[4]{t^{2}-4}}=\frac{t-2}{\sqrt{t^{2}-4}}=\frac{\sqrt{t-2}}{\sqrt{t+2}}=\frac{\sqrt{t^{2}-4}}{t+2}\)

Ответ: \(\frac{\sqrt{t^{2}-4}}{t+2}\)

Решение №16932: \(\frac{\sqrt{\sqrt{3}+2}\sqrt[4]{7-4\sqrt{3}}+\sqrt[3]{\sqrt{x}\left ( x+27 \right )-9x-27}}{\sqrt{x}-2-\sqrt{2-\sqrt{3}}\sqrt[4]{7+4\sqrt{3}}}=\frac{\sqrt{2+\sqrt{3}}\sqrt[4]{\left ( 2-\sqrt{3} \right )^{2}}+\sqrt{x}-3}{\sqrt{x}-2-\sqrt{2-\sqrt{3}}\sqrt[4]{\left ( 2+\sqrt{3} \right )^{2}}}=\frac{\sqrt{2+\sqrt{3}}\sqrt{2-\sqrt{3}}+\sqrt{x}-3}{\sqrt{x}-2-\sqrt{2-\sqrt{3}}\sqrt{2+\sqrt{3}}}=\frac{\sqrt{4-3}+\sqrt{x}-3}{\sqrt{x}-2-\sqrt{4-3}}=\frac{1+\sqrt{x}-3}{\sqrt{x}-2-1}=\frac{\sqrt{x}-2}{\sqrt{x}-3}\)

Ответ: \(\frac{\sqrt{x}-2}{\sqrt{x}-3}\)

Решение №16933: \(\left ( \frac{\sqrt[4]{x^{3}}-\sqrt[4]{x}}{1-\sqrt{x}}+\frac{1+\sqrt{x}}{\sqrt[4]{x}} \right )^{2}\cdot \left ( 1+\frac{2}{\sqrt{x}}+\frac{1}{x} \right )^{-\frac{1}{2}}=\left ( \frac{\sqrt[4]{x}\left ( \sqrt{x}-1 \right )}{1-\sqrt{x}}+\frac{1+\sqrt{x}}{\sqrt[4]{x}} \right )^{2}\cdot \left (\frac{x}{\left ( \sqrt{x}+1 \right )}\right )^{\frac{1}{2}}=\left ( -\sqrt[4]{x} +\frac{1+\sqrt{x}}{\sqrt[4]{x}}\right )^{2}\cdot \frac{\sqrt{x}}{\sqrt{x}+1}=\frac{1}{\sqrt{x}}\cdot \frac{\sqrt{x}}{\sqrt{x}+1}=\frac{1}{\sqrt{x}+1}=\frac{1-\sqrt{x}}{1-x}\)

Ответ: \(\frac{1-\sqrt{x}}{1-x}\)

Решение №16934: \(\frac{\left ( \sqrt[5]{a^{\frac{4}{3}}} \right )^{\frac{3}{2}}}{\left ( \sqrt[5]{a^{4}} \right )^{3}}:\frac{\left ( \sqrt{a\sqrt[3]{a^{2}b}} \right )^{4}}{\left ( \sqrt[4]{a\sqrt{b}} \right )^{6}}=\frac{\left ( a^{\frac{4}{15}} \right )^{\frac{3}{2}}}{\left ( a^{\frac{4}{5}} \right )^{3}}\cdot \frac{\left ( \sqrt{a\cdot a^{\frac{2}{3}}b^{\frac{1}{3}}} \right )^{4}}{\left ( \sqrt[4]{ab^{\frac{1}{2}}} \right )^{6}}=\frac{a^{\frac{2}{5}}}{a^{\frac{12}{5}}}\cdot \frac{\left ( a^{\frac{5}{6}}b^{\frac{1}{6}} \right )^{4}}{a^{\frac{6}{4}}b^{\frac{6}{8}}}=a^{-2}\cdot a^{\frac{10}{3}-\frac{3}{2}}\cdot b^{\frac{2}{3}-\frac{3}{4}}=a^{-2}\cdot a^{\frac{11}{6}}\cdot b^{-\frac{1}{12}}=\frac{1}{\sqrt[12]{a^{2}b}}\)

Ответ: \(\frac{1}{\sqrt[12]{a^{2}b}}\)

Решение №16935: \(\frac{\sqrt[3]{\sqrt{5}-\sqrt{3}}\sqrt[6]{8+2\sqrt{15}}-\sqrt[3]{a}}{\sqrt[3]{\sqrt{2}+\sqrt{12}}\sqrt[6]{8-2\sqrt{15}}-2\sqrt[3]{2a}+\sqrt[3]{a^{2}}}=\frac{\sqrt[3]{5-3}-\sqrt[3]{a}}{\sqrt[3]{2\left ( 5-3 \right )}-2\sqrt[3]{2a}+\sqrt[3]{a^{2}}}=\frac{\sqrt[3]{2}-\sqrt[3]{a}}{\left ( \sqrt[3]{2}-\sqrt[3]{a} \right )}=\frac{1}{\sqrt[3]{2}-\sqrt[3]{a}}\)

Ответ: \(\frac{1}{\sqrt[3]{2}-\sqrt[3]{a}}\)

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: \(\frac{1}{\sqrt[3]{a}+1}\)