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