Решение №17073: \(\frac{1-x^{-2}}{x^{\frac{1}{2}}-x^{-\frac{1}{2}}}-\frac{2}{x^{\frac{3}{2}}}+\frac{x^{-2}-x}{x^{\frac{1}{2}}-x^{-\frac{1}{2}}}=\frac{1-\frac{1}{x^{2}}}{\sqrt{x}-\frac{1}{\sqrt{x}}}+\frac{\frac{1}{x^{2}}-x}{\sqrt{x}-\frac{1}{\sqrt{x}}}-\frac{2}{\sqrt{x^{3}}}=\frac{\left ( 1-x \right )\sqrt{x}}{x-1}-\frac{2}{\sqrt{x^{3}}}=-\sqrt{x}-\frac{2}{\sqrt{x^{3}}}=\frac{-\sqrt{x}\sqrt{x^{3}}-2}{\sqrt{x^{3}}}=\frac{-\sqrt{x^{4}}-2}{\sqrt{x^{3}}}=\frac{-x^{2}-2}{\sqrt{x^{3}}}=-\frac{x^{2}+2}{x\sqrt{x}}=-\frac{\left ( x^{2}+2 \right )\sqrt{x}}{x\sqrt{x\sqrt{x}}}=-\sqrt{x}\left ( 1+\frac{2}{x^{2}} \right )\)

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

Решение №17074: \(x=\sqrt[3]{4+\sqrt{80}}-\sqrt[3]{\sqrt{80}-4}=\sqrt[3]{4+\sqrt{80}}+\sqrt[3]{4-\sqrt{80}};\left ( \sqrt[3]{4+\sqrt{80}}+\sqrt[3]{4-\sqrt{80}} \right )^{3}+12\left ( \sqrt[3]{4+\sqrt{80}}+\sqrt[3]{4-\sqrt{80}} \right )-8=0;\left ( \sqrt[3]{4+\sqrt{80}} \right )^{3}+3\left ( \sqrt[3]{4+\sqrt{80}} \right )^{2}\cdot \sqrt[3]{4-\sqrt{80}}+3\sqrt[3]{4+\sqrt{80}}\cdot \left ( \sqrt[3]{4-\sqrt{80}} \right )^{2}+\left ( \sqrt[3]{4+\sqrt{80}} \right )^{3}+12\left ( \sqrt[3]{4+\sqrt{80}}+\sqrt[3]{4-\sqrt{80}} \right )-8=0;4+\sqrt{80}+3\sqrt[3]{\left ( 4+\sqrt{80} \right )\left ( 4-\sqrt{80} \right )}\left ( \sqrt[3]{4+\sqrt{80}}+\sqrt[3]{4-\sqrt{80}} \right )+4-\sqrt{80}+12\left ( \sqrt[3]{4+\sqrt{80}}+\sqrt[3]{4-\sqrt{80}} \right )-8=0;4+\sqrt{80}-12\left ( \sqrt[3]{4+\sqrt{80}}+\sqrt[3]{4-\sqrt{80}} \right )+4-\sqrt{80}+12\left ( \sqrt[3]{4+\sqrt{80}}+\sqrt[3]{4-\sqrt{80}} \right )-8=0;0=0\)

Ответ: \(0=0\)

Решение №17075: \(\frac{\sqrt{x-2\sqrt{x-1}}}{\sqrt{x-1}-1}=\frac{\sqrt{x-1-2\sqrt{x-1}+1}}{\sqrt{x-1}-1}=\frac{\sqrt{\left ( \sqrt{x-1} \right )^{2}-2\sqrt{x-1}+1}}{\sqrt{x-1}-1}=\frac{\sqrt{\left ( \sqrt{x-1} -1\right )^{2}}}{\sqrt{x-1}-1}=\frac{\left | \sqrt{x-1}-1 \right |}{\sqrt{x-1}-1}=-1;1\)

Ответ: \(-1;1\)

Решение №17076: \(\left ( 4x-1 \right )\left ( \frac{1}{8x}\left ( \sqrt{8x-1}+4x \right )^{-1}-\left ( \sqrt{8x-1}-4x \right )^{-1} \right )^{\frac{1}{2}}=\left ( 4x-1 \right )\sqrt{\frac{\sqrt{8x-1}-4x-\sqrt{8x-1}-4x}{8x\left ( \sqrt{8x-1}+4x \right )\left ( \sqrt{8x-1}-4x \right )}}=\left ( 4x-1 \right )\sqrt{\frac{-1}{8x-1-16x^{2}}}=\left ( 4x-1 \right )\sqrt{\frac{1}{\left ( 4x-1 \right )^{2}}}=\frac{4x-1}{\left | 4x-1 \right |}=-1;1\)

Ответ: \(-1;1\)

Решение №17077: \(\frac{ab^{\frac{2}{3}}-\sqrt[3]{b^{2}}-a+1}{\left ( 1-\sqrt[3]{a} \right )\left ( \left ( \sqrt[3]{a}+1 \right )^{2}-\sqrt[3]{a} \right )\left ( b^{\frac{1}{3}}+1 \right )}+\sqrt[3]{ab}\cdot \left ( \frac{1}{\sqrt[3]{a}}+b^{-\frac{1}{3}} \right )=\frac{a\sqrt[3]{b^{2}}-\sqrt[3]{b^{2}}-a+1}{\left ( 1-\sqrt[3]{a} \right )\left ( \sqrt[3]{a^{2}}+2\sqrt[3]{a}+1-\sqrt[3]{a} \right )\left ( \sqrt[3]{b}+1 \right )}+\sqrt[3]{ab}\left ( \frac{1}{\sqrt[3]{a}}+\frac{1}{\sqrt[3]{b}} \right )=\frac{\left ( a-1 \right )\left ( \sqrt[3]{b^{2}}-1 \right )}{-\left ( a-1 \right )\left ( \sqrt[3]{b}+1 \right )}+\sqrt[3]{b}+\sqrt[3]{a}=-\sqrt[3]{b}+1+\sqrt[3]{b}+\sqrt[3]{a}=1+\sqrt[3]{a}\)

Ответ: \(1+\sqrt[3]{a}\)

Решение №17079: \(\frac{x+\sqrt{x}-\sqrt[4]{12x}+3+\sqrt{3}}{\sqrt{x}+\sqrt{3}-\sqrt[4]{12x}}-\left ( \sqrt{3}+\sqrt[4]{12x} \right )=\frac{\left ( \sqrt[4]{x^{2}}-\sqrt[4]{12x}+\sqrt[4]{3^{2}} \right )+\left ( \sqrt[4]{x^{4}}+\sqrt[4]{3^{4}} \right )}{\sqrt[4]{x^{2}}-\sqrt[4]{12x}+\sqrt[4]{3^{2}}}=\left ( \sqrt[4]{12x}+\sqrt[4]{3^{2}} \right )=1+\frac{\sqrt[4]{x^{4}}-\sqrt[4]{12x^{3}}+\sqrt[4]{144x^{2}}-\sqrt[4]{9x^{2}}}{\sqrt[4]{x^{2}}-\sqrt[4]{12x}+\sqrt[4]{9}}=1+\sqrt[4]{x^{2}}=1+\sqrt{x}\)

Ответ: \(1+\sqrt{x}\)

Решение №17080: \(\frac{2\left ( x^{4}+4x^{2}-12 \right )+x^{4}+11x^{2}+30}{x^{2}+6}=\frac{2\left (x^{2}+6 \right )\left ( x^{2}-2 \right )+\left ( x^{2}+6 \right )\left ( x^{2}+5 \right )}{x^{2}+6}=\frac{\left ( x^{2}+6 \right )\left ( 2\left ( x^{2}-2 \right )+x^{2}+5 \right )}{x^{2}+6}=2x^{2}-4+x^{2}+5=3x^{2}+1=1+3x^{2}\)

Ответ: \(1+3x^{2}\)

Решение №17081: \(\frac{\sqrt{\sqrt[4]{27}+\sqrt{\sqrt{3}-1}}-\sqrt{\sqrt[4]{27}-\sqrt{\sqrt{3}-1}}}{\sqrt{\sqrt[4]{27}-\sqrt{2\sqrt{3}}+1}}=\sqrt{2};\frac{2\sqrt[4]{27}-2\sqrt{\left ( \sqrt[4]{27} \right )^{2}-\left ( \sqrt{\sqrt{3}-1} \right )^{2}}}{\sqrt[4]{27}-\sqrt{2\sqrt{3}+1}}=2;\frac{\sqrt[4]{27}-\sqrt{\sqrt{27}-\sqrt{3}+1}}{\sqrt[4]{27}-\sqrt{2\sqrt{3}+1}}=1;\frac{\sqrt[4]{27}-\sqrt{2\sqrt{3}+1}}{\sqrt[4]{27}-\sqrt{2\sqrt{3}+1}=1;1=1\)

Ответ: \(1=1\)

Решение №17082: \(\sqrt[3]{26+15\sqrt{3}}\cdot \left ( 2-\sqrt{3} \right )=1;\sqrt[3]{26+15\sqrt{3}}\cdot \sqrt[3]{\left ( 2-\sqrt{3} \right )^{3}}=1;\sqrt[3]{26+15\sqrt{3}}\sqrt[3]{8-12\sqrt{3}+18-3\sqrt{3}}=1;\sqrt[3]{\left ( 26+15\sqrt{3} \right )\left ( 26-15\sqrt{3} \right )}=1;\sqrt[3]{26^{2}-\left ( 15\sqrt{3} \right )^{2}}=1;\sqrt[3]{676-675}=1;1=1\)

Ответ: \(1=1\)

Решение №17084: \frac{\sqrt{5-2\sqrt{6}}\left ( 5+2\sqrt{6} \right )\left ( 49-20\sqrt{6} \right )}{\sqrt{27}-3\sqrt{18}+3\sqrt{12}-\sqrt{8}}=1;1=1

Ответ: \(1=1\)

Решение №17085: \(\frac{25*\sqrt[4]{2}+2\sqrt{5}}{\sqrt{250}+5\sqrt[4]{8}}-\sqrt{\frac{\sqrt{2}}{5}+\frac{5}{\sqrt{2}}+2}=-1; \frac{5-\sqrt[4]{5^{2}*2}+\sqrt{2}}{\sqrt[4]{5^{2}}*2}-\frac{5+\sqrt{2}}{\sqrt[4]{5^{2}*2}}=-1;\frac{-\sqrt[4]{5^{4}*2}}{\sqrt[4]{5^{4}*2}}=-1;-1=-1\)

Ответ: \(-1=-1\)

Решение №17088: \(\left ( \left ( 1-x^{2} \right )^{-\frac{1}{2}} +1+\frac{1}{\left ( 1-x^{2} \right )^{-\frac{1}{2}}-1}\right )^{-2}:\left ( 2-x^{2}-2\sqrt{1-x^{2}} \right )=\left ( \frac{1}{\sqrt{1-x^{2}}}+1+\frac{1}{\frac{1}{\sqrt{1-x^{2}}}-1} \right )^{-2}:\left ( 1-2\sqrt{1-x^{2}}+1-x^{2} \right )=\left ( \frac{\left ( 1+\sqrt{1-x^{2}} \right )\left ( 1-\sqrt{1-x^{2}} \right )+\left ( \sqrt{1-x^{2}} \right )^{2}}{\sqrt{1-x^{2}}\left ( 1-\sqrt{1-x^{2}} \right )}\right )^{-2}:\left ( 1-\sqrt{1-x^{2}} \right )=\frac{\left (\sqrt{1-x^{2}}\left ( 1-\sqrt{1-x^{2}} \right ) \right )^{2}}{\left ( 1-\sqrt{1-x^{2}} \right )}=1-x^{2}\)

Ответ: \(1-x^{2}\)

Решение №17089: \(\frac{2\sqrt{\frac{1}{4}\left ( \frac{1}{\sqrt{a}}+\sqrt{a} \right )^{2}}-1}{2\sqrt{\frac{1}{4}\left ( \frac{1}{\sqrt{a}}+\sqrt{a} \right )^{2}-1}-\frac{1}{2}\left ( \sqrt{\frac{1}{a}}-\sqrt{a} \right )}=\frac{\frac{\left | 1-a \right |}{\sqrt{a}}}{\frac{2\left | 1-a \right |-\left ( 1-a \right )}{2\sqrt{a}}}=\frac{\left | 1-a \right |}{\sqrt{a}}\cdot \frac{2\sqrt{a}}{2\left | 1-a \right |-\left ( 1-a \right )}=\frac{2\left | 1-a \right |}{2\left | 1-a \right |-\left ( 1-a \right )}=2;\frac{2}{3}\)

Ответ: \(2;\frac{2}{3}\)

Решение №17090: \(\frac{\left | x^{2}-1 \right |+x^{2}}{2x^{2}-1}-\frac{\left | x-1 \right |}{x-1}=\frac{x^{2}-1+x^{2}}{2x^{2}-1}+\frac{x-1}{x-1};\frac{-x^{2}+1+x^{2}}{2x^{2}-1}+\frac{x-1}{x-1};\frac{x^{2}-1+x^{2}}{2x^{2}-1}-\frac{x-1}{x-1}=2;\frac{2x^{2}}{2x^{2}-1};0\)

Ответ: \(2;\frac{2x^{2}}{2x^{2}-1};0\)

Решение №17091: \(\frac{a^{2}+4}{a\sqrt{\left ( \frac{a^{2}-4}{}2a \right )^{2}+4}}=\frac{a^{2}+4}{a\sqrt{\frac{a^{4}-8a^{2}+16}{4a^{2}}+4}}=\frac{a^{2}+4}{a\sqrt{\frac{a^{4}-8a^{2}+16+16a^{2}}{4a^{2}}}}=\frac{a^{2}+4}{a\sqrt{\left ( \frac{a^{2}+4}{2a} \right )^{2}}}=\frac{a^{2}+4}{\frac{a\left ( a^{2}+4 \right )}{2\left | a \right |}}=\frac{2\left | a \right |}{a}=-2;2\)

Ответ: \(-2;2\)

Решение №17092: \(\frac{x^{2}+4}{x\sqrt{4+\left ( \frac{x^{2}-4}{2x} \right )^{2}}}=\frac{x^{2}+4}{x\sqrt{\frac{x^{4}+8x^{2}+16}{4x^{2}}}}=\frac{x^{2}+4}{x\sqrt{\frac{\left ( x^{2}+4 \right )^{2}}{4x^{2}}}}=\frac{2\left | x \right |}{x}=-2;2\)

Ответ: \(-2;2\)

Решение №17093: \(y=\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}=\sqrt{x-1+2\sqrt{x-1}+1}+\sqrt{x-1-2\sqrt{x-1}+1}=\sqrt{\left ( \sqrt{x-1}+1 \right )^{2}}+\sqrt{\left ( \sqrt{x-1}-1 \right )^{2}}=\sqrt{x-1}+1+\left | \sqrt{x-1}-1 \right |=2;2\sqrt{x-1}\)

Ответ: \(2;2\sqrt{x-1}\)

Решение №17094: \(\frac{14}{\sqrt[4]3{+\sqrt[8]{2}}}=\frac{14}{\sqrt[8]{9}+\sqrt[8]{2}}=\frac{14\left ( \sqrt[8]{9^{7}}-\sqrt[8]{9^{6}*2}-\sqrt[8]{9^{4}*2^{2}}-\sqrt[8]{9^{4}*2^{3}}+\sqrt[8]{9^{3}*2^{4}}-\sqrt[8]{9^{2}*2^{5}}+\sqrt[8]{9*2^{6}}-\sqrt[8]{2^{7}} \right )}{\left ( \sqrt[8]{9}+\sqrt[8]{2} \right )\left ( \sqrt[8]{9^{7}}-\sqrt[8]{9^{6}*2}-\sqrt[8]{9^{4}*2^{2}}-\sqrt[8]{9^{4}*2^{3}}+\sqrt[8]{9^{3}*2^{4}}-\sqrt[8]{9^{2}*2^{5}}+\sqrt[8]{9*2^{6}}-\sqrt[8]{2^{7}} \right )}=\frac{14\left ( \sqrt[8]{9^{3}}-\sqrt[8]{9^{2}*2}+\sqrt[8]{9*2^{2}}-\sqrt[8]{2^{3}} \right )\left ( \sqrt[8]{9^{4}}+\sqrt[8]{2^{4}} \right )}{9-2}=2\left ( \sqrt[4]{3}-\sqrt[8]{2} \right )\left ( \sqrt{3}+\sqrt[4]{2} \right )\left ( 3+\sqrt{2} \right )\)

Ответ: \(2\left ( \sqrt[4]{3}-\sqrt[8]{2} \right )\left ( \sqrt{3}+\sqrt[4]{2} \right )\left ( 3+\sqrt{2} \right )\)

Решение №17095: \(\frac{\left ( \sqrt[8]{x}+\sqrt[8]{y} \right )^{2}+\left ( \sqrt[8]{x}-\sqrt[8]{y} \right )^{2}}{x-\sqrt{xy}}:\frac{\left ( \sqrt[4]{x}+\sqrt[8]{xy}+\sqrt[4]{y} \right )^{2}\left ( \sqrt[4]{x}-\sqrt[8]{xy}+\sqrt[4]{y} \right )^{2}}{\sqrt[4]{x^{3}y}-y}=\frac{2}{\sqrt[8]{x^{4}}\left ( \sqrt[8]{x^{2}}-\sqrt[8]{y^{2}} \right )}:\frac{\sqrt[8]{x^{4}}+\sqrt[8]{x^{2}y^{2}}+\sqrt[8]{y^{4}}}{\sqrt[8]{y^{2}}\left ( \sqrt[8]{x^{2}}-\sqrt[8]{y^{2}} \right )\left (\sqrt[8]{x^{4}}+\sqrt[8]{x^{2}y^{2}}+\sqrt[8]{y^{4}} \right )}=\frac{2}{\sqrt[8]{x^{4}}\left ( \sqrt[8]{x^{2}}-\sqrt[8]{y^{2}} \right )}:\frac{1}{\sqrt[8]{y^{2}}\left ( \sqrt[8]{x^{2}}-\sqrt[8]{y^{2}} \right )}=\frac{2\sqrt[8]{y^{2}}\left ( \sqrt[8]{x^{2}}-\sqrt[8]{y^{2}} \right )}{\sqrt[8]{x^{4}}\left ( \sqrt[8]{x^{2}}-\sqrt[8]{y^{2}} \right )}=\frac{2\sqrt[8]{y^{2}}}{\sqrt[8]{x^{4}}}=2\sqrt[4]{\frac{y}{x^{2}}}\)

Ответ: \(2\sqrt[4]{\frac{y}{x^{2}}}\)

Решение №17097: \(\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}}=\sqrt{x-2+2\sqrt{2\left ( x-2 \right )}+2}+\sqrt{x-2-2\sqrt{2\left ( x-2 \right )}+2}=\sqrt{\left ( \sqrt{x-2}+\sqrt{2} \right )^{2}}+\sqrt{\left ( \sqrt{x-2}-\sqrt{2} \right )^{2}}=\sqrt{x-2}+\sqrt{2}+\left | \sqrt{x-2}-\sqrt{2} \right |=2\sqrt{2};2\sqrt{x-2}\)

Ответ: \(2\sqrt{2};2\sqrt{x-2}\)

Решение №17098: \(\sqrt[3]{45+29\sqrt{2}}-\sqrt[3]{45-29\sqrt{2}}=2\sqrt{2};\sqrt[3]{27+27\sqrt{2}+18+2\sqrt{2}}-\sqrt[3]{27-27\sqrt{2}+18-2\sqrt{2}}=3+\sqrt{2}-3+\sqrt{2}=2\sqrt{2};2\sqrt{2}=2\sqrt{2}\)

Ответ: \(2\sqrt{2}=2\sqrt{2}\)

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: \(-2a;2a\)