Решение №17057: \(\frac{x+\sqrt{3}}{\sqrt{x}+\sqrt{x+\sqrt{3}}}+\frac{x-\sqrt{3}}{\sqrt{x}-\sqrt{x-\sqrt{3}}}=\frac{x\sqrt{x}-x\sqrt{x+\sqrt{3}}+\sqrt{3x}-\sqrt{3\left ( x+\sqrt{3} \right )}}{x-x-\sqrt{3}}+\frac{x\sqrt{x}+x\sqrt{x+\sqrt{3}}-\sqrt{3x}-\sqrt{3\left ( x+\sqrt{3} \right )}}{x-x+\sqrt{3}}=\frac{-x\sqrt{x}+x\sqrt{x+\sqrt{3}}-\sqrt{3x}+\sqrt{3\left ( x+\sqrt{3} \right )}+x\sqrt{x}+x\sqrt{x-\sqrt{3}}-\sqrt{3x}-\sqrt{3\left ( x-\sqrt{3} \right )}}{\sqrt{3}}=\frac{\sqrt{\left ( x+\sqrt{3} \right )^{3}}+\sqrt{\left ( x-\sqrt{3} \right )^{3}}}{\sqrt{3}}-2\sqrt{x}=\sqrt{2+\sqrt{3}+2+2-\sqrt{3}}\sqrt{3}-2\sqrt{2}=\sqrt{6}\sqrt{3}-2\sqrt{2}=3\sqrt{2}-2\sqrt{2}=\sqrt{2}\)

Ответ: \(\sqrt{2}\)

Решение №17058: \(\frac{\sqrt{11+\sqrt{3}}}{\sqrt{59}}\cdot \sqrt{4+\sqrt{5+\sqrt{3}}}\sqrt{3+\sqrt{5+\sqrt{5+\sqrt{3}}}}\sqrt{3-\sqrt{5+\sqrt{5+\sqrt{3}}}}=\frac{\sqrt{11+\sqrt{3}}}{\sqrt{59}}\cdot \sqrt{4+\sqrt{5+\sqrt{3}}}\sqrt{4-\sqrt{5+\sqrt{3}}}=\frac{\sqrt{11+\sqrt{3}}}{\sqrt{59}}\sqrt{4^{2}-\left ( \sqrt{5}+\sqrt{3} \right )^{2}}=\frac{\sqrt{11+\sqrt{3}}}{\sqrt{59}}\sqrt{11-\sqrt{3}}=\frac{\sqrt{11^{2}-\sqrt{3}^{2}}}{\sqrt{59}}=\frac{\sqrt{118}}{\sqrt{59}}=\sqrt{\frac{118}{59}}=\sqrt{2}\)

Ответ: \(\sqrt{2}\)

Решение №17059: \(\left ( \frac{3}{\sqrt[4]{64}-\sqrt[3]{25}}+\frac{\sqrt[3]{40}}{\sqrt[3]{8}+\sqrt[3]{5}}-\frac{10}{\sqrt[3]{25}} \right ):\left ( \sqrt[6]{8}+\sqrt[6]{5} \right )+\sqrt[6]{5}=\sqrt{2};\left ( \frac{3\left ( 16+4\sqrt[3]{5^{2}+2\sqrt[3]{5}} \right )}{39} +\frac{2\sqrt[3]{5}\left ( 4-2\sqrt[3]{5}+\sqrt[3]{5^{2}} \right )}{13}-2\sqrt[3]{5}\right )\cdot \frac{1}{\sqrt[6]{8}+\sqrt[6]{5}}+\sqrt[6]{5}=\frac{16+4\sqrt[3]{5^{2}}+2\sqrt[3]{5}+8\sqrt[3]{5}-4\sqrt[3]{5}+10-26\sqrt[3]{5}}{13}\cdot \frac{1}{\sqrt[6]{8}+\sqrt[6]{5}}+\sqrt[6]{5}=\frac{2-\sqrt[3]{5}}{\sqrt[6]{8}+\sqrt[6]{5}}+\sqrt[6]{5}=\frac{2-\sqrt[3]{5}+\sqrt[6]{40}+\sqrt[6]{5^{2}}}{\sqrt[6]{8}+\sqrt[6]{5}}=\frac{2-\sqrt[3]{5}+\sqrt{2}\sqrt[6]{5}+\sqrt[3]{5}}{\sqrt[6]{8}+\sqrt[6]{5}}=\frac{2+\sqrt{2}\sqrt[6]{5}}{\sqrt{2}+\sqrt[6]{5}}=\frac{\sqrt{2}\left ( \sqrt{2}+\sqrt[6]{5} \right )}{\sqrt{2}+\sqrt[6]{5}}=\sqrt{2}\)

Ответ: \(\sqrt{2}\)

Решение №17060: \(\left ( \sqrt{\sqrt{m}-\sqrt{\frac{m^{2}-9}{m}}}+\sqrt{\sqrt{m}+\sqrt{\frac{m^{2}-9}{m}}} \right )\sqrt[4]{\frac{m^{2}}{4}}=\left ( 2\sqrt{m}+2\sqrt{m-\frac{m^{2}-9}{m}} \right )\sqrt{\frac{m}{2}}=\left ( 2\sqrt{m}+\frac{6}{\sqrt{m}} \right )\cdot \frac{\sqrt{m}}{\sqrt{2}}=m\sqrt{2}+\frac{6}{\sqrt{2}}=\sqrt{2}\left ( m+3 \right )\)

Ответ: \(\sqrt{2}\left ( m+3 \right )\)

Решение №17061: \(\frac{\sqrt{21+8\sqrt{5}}{4+\sqrt{5}}}\sqrt{9-4\sqrt{5}}=\sqrt{5}-2;\frac{\sqrt{16+8\sqrt{5}+5}}{4+\sqrt{5}}\sqrt{5-4\sqrt{5}+4}=\sqrt{5}-2;\frac{4+\sqrt{5}}{4+\sqrt{5}}\left ( \sqrt{5}-2 \right )=\sqrt{5}-2;\sqrt{5}-2=\sqrt{5}-2\)

Ответ: \(\sqrt{5}-2=\sqrt{5}-2\)

Решение №17062: \(\frac{1}{\sqrt{7}-\sqrt{6}}=\frac{3}{\sqrt{6}-\sqrt{3}}+\frac{4}{\sqrt{7}+\sqrt{3}};\frac{\sqrt{7}+\sqrt{6}}{\left ( \sqrt{7}-\sqrt{6} \right )\left ( \sqrt{7}+\sqrt{6} \right )}=\frac{3\left ( \sqrt{6}+\sqrt{3} \right )}{\left ( \sqrt{6}-\sqrt{3} \right )\left ( \sqrt{6}+\sqrt{3} \right )}+\frac{4\left ( \sqrt{7}-\sqrt{3} \right )}{\left ( \sqrt{7}+\sqrt{3} \right )\left ( \sqrt{7}-\sqrt{3} \right )}; \frac{\sqrt{7}+\sqrt{6}}{7-6}=\frac{3\left ( \sqrt{6}+\sqrt{3} \right )}{6-3}+\frac{4\left ( \sqrt{7}-\sqrt{3} \right )}{7-3}; \sqrt{7}+\sqrt{6}=\sqrt{6}+\sqrt{3}+\sqrt{7}-\sqrt{3}; \sqrt{6}=\sqrt{6}\)

Ответ: \(\sqrt{6}=\sqrt{6}\)

Решение №17063: \(\sqrt[4]{6x\left ( 5+2\sqrt{6} \right )}\cdot \sqrt{3\sqrt{2x}-2\sqrt{3x}}=\sqrt[4]{6x\left ( 5+2\sqrt{6} \right )}\cdot \sqrt{\sqrt{6x}\left ( \sqrt{3}-\sqrt{2} \right )}=\sqrt[4]{6x\left ( 5+2\sqrt{6} \right )}\cdot \sqrt[4]{\left ( \sqrt{6x}\left ( \sqrt{3}-\sqrt{2} \right ) \right )^{2}}=\sqrt[4]{6x\left ( 5+2\sqrt{6} \right )}\cdot \sqrt[4]{6x\left ( 5-2\sqrt{6} \right )}=\sqrt[4]{6x\left ( 5+2\sqrt{6} \right )6x\left ( 5-2\sqrt{6} \right )}=\sqrt[4]{36x^{2}\left ( 25-24 \right )}=\sqrt[4]{36x^{2}}=\sqrt{6x}\)

Ответ: \(\sqrt{6x}\)

Решение №17064: \(\frac{3}{\sqrt{5}-\sqrt{2}}+\frac{5}{\sqrt{7}+\sqrt{2}}=\frac{2}{\sqrt{7}-\sqrt{5}};\frac{3\left ( \sqrt{5}+\sqrt{2} \right )}{\left ( \sqrt{5}-\sqrt{2} \right )\left ( \sqrt{5}+\sqrt{2} \right )}+\frac{5\left ( \sqrt{7}-\sqrt{2} \right )}{\left ( \sqrt{7}+\sqrt{2} \right )\left ( \sqrt{7}-\sqrt{2} \right )}=\frac{2\left ( \sqrt{7}+\sqrt{5} \right )}{\left ( \sqrt{7}-\sqrt{5} \right )\left ( \sqrt{7}+\sqrt{5} \right )}; \frac{3\left ( \sqrt{5}+\sqrt{2} \right )}{5-2}+\frac{5\left ( \sqrt{7}-\sqrt{2} \right )}{7-2}=\frac{2\left ( \sqrt{7}+\sqrt{5} \right )}{7-5}; \sqrt{5}+\sqrt{2}+\sqrt{7}-\sqrt{2}=\sqrt{7}+\sqrt{5}; \sqrt{7}+\sqrt{5}=\sqrt{7}+\sqrt{5}\)

Ответ: \(\sqrt{7}+\sqrt{5}=\sqrt{7}+\sqrt{5}\)

Решение №17065: \(1-\frac{\frac{1}{\sqrt{a-1}}-\sqrt{a+1}}{\frac{1}{\sqrt{a+1}}-\frac{1}{\sqrt{a-1}}}:\frac{\sqrt{a+1}\sqrt{a^{2}-1}}{\left ( a-1 \right )\sqrt{a+1}-\left ( a+1 \right )\sqrt{a-1}}=1-\frac{\frac{1-\sqrt{a^{2}-1}}{\sqrt{a-1}}}{\frac{\sqrt{a-1}-\sqrt{a+1}}{\sqrt{\left ( a+1 \right )\left ( a-1 \right )}}}:\frac{\sqrt{a+1}\sqrt{\left ( a-1 \right )\left ( a+1 \right) }}{\left ( \sqrt{a-1}-\sqrt{a+1} \right )}=1-\frac{\left ( 1-\sqrt{a^{2}-1} \right )\sqrt{\left ( a+1 \right )\left ( a-1 \right )}}{\sqrt{a-1}\left ( \sqrt{a-1}-\sqrt{a+1} \right )}\cdot \frac{\sqrt{a-1}-\sqrt{a+1}}{\sqrt{a+1}}=1-1+\sqrt{a^{2}-1}=\sqrt{a^{2}-1}\)

Ответ: \(\sqrt{a^{2}-1}\)

Решение №17066: \(\frac{\frac{1}{\sqrt{a-1}}-\sqrt{a+1}}{\frac{1}{\sqrt{a+1}}-\frac{1}{\sqrt{a-1}}}:\frac{\sqrt{a+1}}{\left ( a-1 \right )\sqrt{a+1}-\left ( a+1 \right )\sqrt{a-1}}-\left ( 1-a^{2} \right )=\frac{1-\sqrt{\left ( a-1 \right )\left ( a+1 \right )}}{\sqrt{a-1}}\cdot \frac{\sqrt{\left ( a-1 \right )\left ( a+1 \right )}}{\sqrt{a-1}-\sqrt{a+1}}\cdot \frac{\sqrt{\left ( a-1 \right )\left ( a+1 \right )}\left ( \sqrt{a-1}-\sqrt{a+1} \right )}{\sqrt{a+1}}-\left ( 1-a^{2} \right )=\left ( 1-\sqrt{\left ( a-1 \right )\left ( a+1 \right )} \right )\sqrt{\left ( a-1 \right )\left ( a+1 \right )}-\left ( 1-a^{2} \right )=\sqrt{a^{2}-1}-a^{2}+1-1+a^{2}=\sqrt{a^{2}-1}\)

Ответ: \(\sqrt{a^{2}-1}\)