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 №17096: \(5\sqrt{48\sqrt[3]{\frac{2}{3}}}+\sqrt{32\sqrt[3]{\frac{9}{4}}}-11\sqrt[3]{12\sqrt{8}}=5\sqrt{16\cdot 3\sqrt[3]{\frac{2}{3}}}+\sqrt{16\cdot 2\sqrt[3]{\frac{9}{4}}}-11\sqrt[3]{12\cdot 2\sqrt{2}}=20\sqrt{\sqrt[3]{18}}+4\sqrt{\sqrt[3]{18}}-22\sqrt[3]{\sqrt{9\cdot 2}}=24\sqrt{\sqrt[3]{18}}-22\sqrt{\sqrt[3]{18}}=2\sqrt[6]{18}\)

Ответ: \(2\sqrt[6]{18}\)

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: \(2\sqrt{2}=2\sqrt{2}\)