Решение №17038: \(\frac{\sqrt[3]{\sqrt{3}+\sqrt{6}}\sqrt[6]{9-6\sqrt{2}}-\sqrt[6]{18}}{\sqrt[6]{2}-1}=-\sqrt[3]{3}; \frac{\sqrt[3]{\left ( \sqrt{3}+\sqrt{6} \right )^{2}}\sqrt[6]{9-6\sqrt{2}}-\sqrt[6]{18}}{\sqrt[6]{2}-1}=-\sqrt[3]{3};\frac{\sqrt[6]{9^{2}-\left ( 6\sqrt{2} \right )^{2}-\sqrt[6]{18}}}{\sqrt[6]{2}-1}=-\sqrt[3]{3}; \frac{\sqrt[6]{3^{2}}\left ( 1-\sqrt[6]{2} \right )}{\sqrt[6]{2}-1}=-\sqrt[3]{3};-\sqrt[3]{3}=-\sqrt[3]{3}\)

Ответ: \(-\sqrt[3]{3}=-\sqrt[3]{3}\)

Решение №17039: \(\left ( \frac{4-2x+x^{2}}{4-2x}+\frac{6x^{2}+8+12x}{4-x^{2}}-\frac{x^{2}+2x+4}{2x+4} \right )^{-\frac{1}{3}}\cdot \left ( x+2 \right )=\left ( -\frac{x^{2}-2x+4}{2\left ( x-2 \right )}-\frac{6x^{2}+12x+8}{\left ( x-2 \right )\left ( x+2 \right )}-\frac{x^{2}+2x+4}{2\left ( x+2 \right )} \right )^{-\frac{1}{3}}\cdot \left ( x+2 \right )=-\left ( \frac{2\left ( x^{3}+6x^{2}+12x+8 \right )}{2\left ( x-2 \right )\left ( x+2 \right )} \right )^{-\frac{1}{3}}\cdot \left ( x+2 \right )=-\sqrt[3]{\frac{\left ( x-2 \right )\left ( x+2 \right )^{3}}{\left ( x+2 \right )^{2}}}=-\sqrt[3]{x^{2}-4}=\sqrt[3]{4-x^{2}}\)

Ответ: \(\sqrt[3]{4-x^{2}}\)

Решение №17040: \(\frac{\sqrt[3]{ab}\left ( \sqrt[3]{b^{2}}-\sqrt[3]{a^{2}} \right )+\sqrt[3]{a^{4}}-\sqrt[3]{b^{4}}}{\sqrt[3]{a^{4}}+\sqrt[3]{a^{2}b^{2}}-\sqrt[3]{a^{3}b}}\cdot \sqrt[3]{a^{2}}=\frac{\left ( \sqrt[3]{a^{2}}-\sqrt[3]{b^{2}} \right )\left ( \sqrt[3]{a^{2}-\sqrt[3]{ab}+\sqrt[3]{b^{2}}} \right )}{\sqrt[3]{a^{2}}-\sqrt[3]{ab}+\sqrt[3]{b^{2}}}=\sqrt[3]{a^{2}}-\sqrt[3]{b^{2}}\)

Ответ: \(\sqrt[3]{a^{2}}-\sqrt[3]{b^{2}}\)

Решение №17041: \(\left ( \frac{\sqrt[3]{x+y}}{\sqrt[3]{x-y}}+\frac{\sqrt[3]{x-y}}{\sqrt[3]{x+y}}-2\right ):\left ( \frac{1}{\sqrt[3]{x-y}}-\frac{1}{\sqrt[3]{x+y}} \right )=\frac{\left ( \sqrt[3]{x+y} \right )^{2}-2\sqrt[3]{\left ( x+y \right )\left ( x-y \right )}+\left ( \sqrt[3]{x-y} \right )^{2}}{\sqrt[3]{x^{2}-y^{2}}}:\frac{\sqrt[3]{x+y}-\sqrt[3]{x-y}}{\sqrt[3]{x^{2}-y^{2}}}=\sqrt[3]{x+y}-\sqrt[3]{x-y}\)

Ответ: \(\sqrt[3]{x+y}-\sqrt[3]{x-y}\)

Решение №17043: \(\left ( \frac{\sqrt[4]{8}+2}{\sqrt[4]{2}+\sqrt[3]{2}}-\sqrt[3]{4} \right ):\left (\frac{\sqrt[4]{8}-2}{\sqrt[4]{2}-\sqrt[3]{2}} -3\sqrt[12]{128} \right )^{\frac{1}{2}}=\frac{\sqrt[12]{2^{9}}+\sqrt[12]{2^{12}}}{\sqrt[12]{2^{3}}+\sqrt[12]{2^{4}}}-\sqrt[12]{2^{8}}:\sqrt{\frac{\sqrt[12]{2^{9}}+\sqrt[12]{2^{12}}}{\sqrt[12]{2^{3}}+\sqrt[12]{2^{4}}}-3\sqrt[12]{2^{7}}}=\sqrt[12]{2^{6}}-\sqrt[12]{2^{7}}:\sqrt{\left ( \sqrt[12]{2^{6}}-\sqrt[12]{2^{7}} \right )^{2}}=\frac{\sqrt{2}\left ( 1-\sqrt[12]{2} \right )}{\sqrt[4]{2}\left ( \sqrt[12]{2}-1 \right )}=-\sqrt[4]{2}\)

Ответ: \(-\sqrt[4]{2}\)

Решение №17044: \(\left ( \frac{\sqrt[4]{x^{3}}-y}{\sqrt[4]{x}-\sqrt[3]{y}}-3\sqrt[12]{x^{3}y^{4}} \right )^{-\frac{1}{2}}\left ( \frac{\sqrt[4]{x^{3}}+y}{\sqrt[4]{x}+\sqrt[3]{y}}-\sqrt[3]{y^{2}} \right )=\left ( \sqrt[4]{x^{2}}-2\sqrt[4]{x\sqrt[3]{y}}+\sqrt[3]{y^{2}} \right )^{-\frac{1}{2}}\left ( \sqrt[4]{x^{2}}-\sqrt[4]{x\sqrt[3]{y}} \right )=-\sqrt[4]{x};\sqrt[4]{x}\)

Ответ: \(-\sqrt[4]{x};\sqrt[4]{x}\)

Решение №17045: \(\frac{\sqrt[3]{x+\sqrt{2-x^{2}}}\sqrt[6]{1-x\sqrt{2-x^{2}}}}{\sqrt[3]{1-x^{2}}}=\frac{\sqrt[6]{\left ( x+\sqrt{2-x^{2}} \right )^{2}}\sqrt[6]{1-x\sqrt{2-x^{2}}}}{\sqrt[3]{1-x^{2}}}=\frac{\sqrt[6]{\left ( 1+x\sqrt{2-x^{2}} \right )\left ( 1-x\sqrt{2-x^{2}} \right )}}{\sqrt[3]{1-x^{2}}}=\frac{\sqrt[6]{2\left ( 1-x^{2}\left ( 2-x^{2} \right )^{2} \right )}}{\sqrt[3]{1-x^{2}}}=\frac{\sqrt[6]{2\left ( 1-x^{2} \right )^{2}}}{\sqrt[3]{1-x^{2}}}=\frac{\sqrt[6]{2}\left | \sqrt[3]{1-x^{2}} \right |}{\sqrt[3]{1-x^{2}}}=-\sqrt[6]{2};\sqrt[6]{2}\)

Ответ: \(-\sqrt[6]{2};\sqrt[6]{2}\)

Решение №17046: \(\frac{\sqrt[3]{2a+2\sqrt{a^{2}-1}}}{\left ( \frac{\sqrt{a-1}}{\sqrt{a+1}}+\frac{\sqrt{a+1}}{\sqrt{a-1}}+2 \right )^{\frac{1}{3}}}=\frac{\sqrt[3]{\left ( \sqrt{a-1} \right )^{2}+2\sqrt{\left ( a-1 \right )\left ( a+1 \right )}+\left ( \sqrt{a+1} \right )^{2}}}{\sqrt[3]{\frac{\left ( \sqrt{a-1}+\sqrt{a+1} \right )^{2}}{\sqrt{a^{2}-1}}}}=\sqrt[3]{\left ( \sqrt{a-1}+\sqrt{a+1} \right )^{2}}\cdot \frac{\sqrt[6]{a^{2}-1}}{\sqrt[3]{\left ( \sqrt{a-1} +\sqrt{a+1}\right )^{2}}}=\sqrt[6]{a^{2}-1}\)

Ответ: \(\sqrt[6]{a^{2}-1}\)

Решение №17047: \(\left ( \frac{\sqrt[3]{mn^{2}}+\sqrt[3]{m^{2}n}}{\sqrt[3]{m^{2}}+2\sqrt[3]{mn}+\sqrt[3]{n^{2}}}-2\sqrt[3]{n}+\frac{m-n}{\sqrt[3]{m^{2}}-\sqrt[3]{n^{2}}} \right ):\left ( \sqrt[6]{m}+\sqrt[6]{n} \right )=\frac{\sqrt[3]{m}-\sqrt[3]{n}}{\sqrt[6]{m}+\sqrt[6]{n}}=\frac{\left ( \sqrt[6]{m}+\sqrt[6]{n} \right )\left ( \sqrt[6]{m}-\sqrt[6]{n} \right )}{\sqrt[6]{m}+\sqrt[6]{n}}=\sqrt[6]{m}-\sqrt[6]{n}\)

Ответ: \(\sqrt[6]{m}-\sqrt[6]{n}\)

Решение №17049: \(\sqrt[n]{y^{\frac{2n}{m-n}}}:\sqrt[m]{y^{\frac{\left ( m-n \right )^{2}+4mn}{m^{2}-n^{2}}}}=y^{\frac{2n}{n\left ( m-n \right )}}:y^{\frac{m^{2}-2mn+n^{2}+4mn}{m\left ( m+n \right )\left ( m-n \right )}}=y^{\frac{2}{m-n}}:y^{\frac{\left ( m+n \right )^{2}}{m\left ( m+n \right )\left ( m-n \right )}}=y^{\frac{2}{m-n}-\frac{m+n}{m\left ( m-n \right )}}=y^{\frac{2m-m-n}{m\left ( m-n \right )}}=y^{\frac{1}{m}}=\sqrt[m]{y}\)

Ответ: \(\sqrt[m]{y}\)

Решение №17052: \(\frac{t^{2}-t-6-\left ( t+3 \right )\sqrt{t^{2}-4}}{t^{2}+t-6-\left ( t-3 \right )\sqrt{t^{2}-4}}=\frac{\left ( t^{2}-4 \right )-\left ( t+2 \right )-\left ( t+3 \right )\sqrt{t^{2}-4}}{\left ( t^{2}-4 \right )+\left ( t-2 \right )-\left ( t-3 \right )\sqrt{t^{2}-4}}=\frac{\left ( t-3 \right )\left ( t+2 \right )-\left ( t+3 \right )\sqrt{\left ( t-2 \right )\left ( t+2 \right )}}{\left ( t+3 \right )\left ( t-2 \right )-\left ( t-3 \right )\sqrt{\left ( t-2 \right )\left ( t+2 \right )}}=-\frac{\sqrt{t+2}}{\sqrt{t-2}}=-\sqrt{\frac{t+2}{t-2}} \)

Ответ: \(-\sqrt{\frac{t+2}{t-2}}\)

Решение №17053: \(\frac{x^{2}+2x-3+\left ( x+1 \right )\sqrt{x^{2}-9}}{x^{2}-2x-3+\left ( x-1 \right )\sqrt{x^{2}-9}}=\frac{\left ( x^{2}-1 \right )+2\left ( x-1 \right )+\left ( x+1 \right )\sqrt{x^{2}-9}}{\left ( x^{2}-1 \right )-2\left ( x+1 \right )+\left ( x-1 \right )\sqrt{x^{2}-9}}=\frac{\left ( x+3 \right )\left ( x-1 \right )+\left ( x+1 \right )\sqrt{\left ( x-3 \right )\left ( x+3 \right )}}{\left ( x-3 \right )\left ( x+1 \right )+\left ( x-1 \right )\sqrt{\left ( x-3 \right )\left ( x+3 \right )}}=\frac{\sqrt{x+3}}{\sqrt{x-3}}=\sqrt{\frac{x+3}{x-3}}\)

Ответ: \(\sqrt{\frac{x+3}{x-3}}\)

Решение №17054: \(\frac{\left ( \left ( x+2 \right )^{-\frac{1}{2}} +\left ( x-2 \right )^{-\frac{1}{2}}\right )^{-1}+\left ( \left ( x+2 \right )^{-\frac{1}{2}}-\left ( x*2 \right )^{-\frac{1}{2}} \right )^{-1}}{\left ( \left ( x+2 \right )^{-\frac{1}{2}}+\left ( x-2 \right )^{-\frac{1}{2}} \right )^{-1}-\left ( \left ( x+2 \right )^{-\frac{1}{2}}-\left ( x-2 \right )^{-\frac{1}{2}} \right )^{-1}}=\frac{\left ( \frac{1}{\sqrt{x+2}}+\frac{1}{\sqrt{x-2}} \right )^{-1}+ \left ( \frac{1}{\sqrt{x+2}}-\frac{1}{\sqrt{x-2}} \right )^{-1}}{ \left ( \frac{1}{\sqrt{x+2}}+\frac{1}{\sqrt{x-2}} \right )^{-1} -\left ( \frac{1}{\sqrt{x+2}}-\frac{1}{\sqrt{x-2}} \right )^{-1}}=\frac{\frac{\sqrt{x-2}-\sqrt{x+2}+\sqrt{x-2}+\sqrt{x+2}}{\left ( \sqrt{x-2}+\sqrt{x+2} \right )\left ( \sqrt{x-2}-\sqrt{x+2} \right )}}{\frac{\sqrt{x-2}+\sqrt{x+2}-\sqrt{x-2}+\sqrt{x+2}}{\left ( \sqrt{x-2}+\sqrt{x+2} \right )\left ( \sqrt{x-2}-\sqrt{x+2} \right )}}=-\frac{\sqrt{x-2}}{\sqrt{x+2}}=-\sqrt{\frac{x-2}{x+2}}\)

Ответ: \(-\sqrt{\frac{x-2}{x+2}}\)

Решение №17055: \(\sqrt{\frac{p^{2}-q\sqrt{p}}{\sqrt{p}-\sqrt[3]{q}}+p\sqrt[3]{q}}\left ( p+\sqrt[6]{p^{3}q^{2}} \right )^{-\frac{1}{2}}=\sqrt{\sqrt{p}\left ( \sqrt{p^{2}}+\sqrt{p}\sqrt[3]{q}+\sqrt[3]{q^{2}}\right )+p\sqrt[3]{q}}\cdot \frac{1}{\sqrt{\sqrt{p}\left ( \sqrt{p}+\sqrt[3]{q} \right )}}=\sqrt{\frac{\left ( \sqrt{p}+\sqrt[3]{q} \right )^{2}}{\sqrt{p}+\sqrt[3]{q}}}=\sqrt{\sqrt{p}+\sqrt[3]{q}}\)

Ответ: \(\sqrt{\sqrt{p}+\sqrt[3]{q}}\)

Решение №17056: \(\left ( \sqrt{1-x^{2}}+1 \right ):\left ( \frac{1}{\sqrt{1+x}}+\sqrt{1-x} \right )=\left ( \sqrt{1-x^{2}}+1 \right ):\frac{1+\sqrt{1-x^{2}}}{\sqrt{1+x}}=\frac{\left ( \sqrt{1-x^{2}}+1 \right )\sqrt{1+x}}{1+\sqrt{1-x^{2}}}=\sqrt{1+x}\)

Ответ: \(\sqrt{1+x}\)

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: \(\sqrt{a^{2}-1}\)

Решение №17067: \(\left ( \frac{1}{\sqrt{a}+\sqrt{a+1}} +\frac{1}{\sqrt{a}-\sqrt{a-1}}\right ):\left ( 1+\sqrt{\frac{a+1}{a-1}} \right )=\frac{\sqrt{a}-\sqrt{a-1}+\sqrt{a}+\sqrt{a+1}}{\left ( \sqrt{a}+\sqrt{a+1} \right )\left (\sqrt{a}-\sqrt{a-1} \right )}:\frac{\sqrt{a+1}+\sqrt{a-1}}{\sqrt{a-1}}=\frac{2\sqrt{a}+\sqrt{a+1}-\sqrt{a-1}}{a+\sqrt{a\left ( a+1 \right )}-\sqrt{a\left ( a-1 \right )}-\sqrt{\left ( a-1 \right )\left ( a+1 \right )}}:\frac{\sqrt{a+1}+\sqrt{a-1}}{\sqrt{a-1}}=\frac{\sqrt{a-1}\left ( 2\sqrt{a}+\sqrt{a+1}-\sqrt{a-1} \right )}{2\sqrt{a}+\sqrt{a+1}-\sqrt{a-1}}=\sqrt{a-1}\)

Ответ: \(\sqrt{a-1}\)

Решение №17068: \(\frac{\sqrt{3}\left ( a-b^{2} \right )+\sqrt{3}b\sqrt[3]{8b^{3}}}{\sqrt{2\left ( a-b^{2} \right )^{2}+\left ( 2b\sqrt{2a} \right )^{2}}}\cdot \frac{\sqrt{2a}-\sqrt{2c}}{\sqrt{\frac{3}{a}-\sqrt{\frac{3}{c}}}}=\frac{\sqrt{3}\left ( a-b^{2}+2b^{2} \right )}{\sqrt{2}\sqrt{a^{2}-2ab^{2}+b^{4}+4ab^{2}}}\cdot \frac{\sqrt{2}\left ( \sqrt{a}-\sqrt{c} \right )\sqrt{ac}}{\sqrt{3}\left ( \sqrt{c}-\sqrt{a} \right )}=\frac{a+b^{2}}{\sqrt{a^{2}+2ab^{2}+b}}\cdot \frac{-\sqrt{ac}}{1}=\frac{-\left ( a+b^{2} \right )\sqrt{ac}}{\sqrt{\left ( a+b^{2} \right )^{2}}}=-\sqrt{ac}\)

Ответ: \(-\sqrt{ac}\)

Решение №17069: \(\frac{\left ( x\sqrt[4]{x}-\sqrt{xy}\left ( \sqrt[4]{x}-\sqrt[4]{y} \right )-y\sqrt[4]{y} \right )\left ( x+y+\sqrt{xy} \right )}{\left ( \sqrt[4]{x}+\sqrt[4]{y} \right )\left ( \left ( \sqrt[4]{x}-\sqrt[4]{y} \right )^{2} +\sqrt[4]{xy}\right )}=\frac{\left ( \sqrt[4]{x^{2}}-\sqrt[4]{y^{2}} \right )\left ( \sqrt[4]{x^{3}}+\sqrt[4]{y^{3}} \right )\left ( \sqrt[4]{x^{4}}+\sqrt[4]{x^{2}y^{2}}+\sqrt[4]{y^{4}} \right )}{\sqrt[4]{x^{3}}+\sqrt[4]{y^{3}}}=\left ( \sqrt[4]{x^{2}}-\sqrt[4]{y^{2}} \right )\left ( \sqrt[4]{x^{4}}+\sqrt[4]{x^{2}y^{2}}+\sqrt[4]{y^{4}} \right )=\left ( \sqrt{x}-\sqrt{y} \right )\left ( \sqrt{x^{2}}+\sqrt{xy}+\sqrt{y^{2}} \right )=\sqrt{x^{3}}-\sqrt{y^{3}}\)

Ответ: \(\sqrt{x^{3}}-\sqrt{y^{3}}\)

Решение №17070: \(\frac{\sqrt{\left ( x+2 \right )^{2}-8x}}{\sqrt{x}-\frac{2}{\sqrt{x}}}=\frac{\sqrt{x^{2}+4x+4-8x}}{\frac{\sqrt{x}^{2}-2}{\sqrt{x}}}=\frac{\sqrt{x}\sqrt{x^{2}-4x-4}}{x-2}=\frac{\sqrt{x}\sqrt{\left ( x^{2}-2 \right )^{2}}}{x-2}=\frac{\sqrt{x}\left | x-2 \right |}{x-2}\)

Ответ: \(\sqrt{x},-\sqrt{x}\)

Решение №17072: \(\frac{x-1}{x^{\frac{3}{4}}+x^{\frac{1}{2}}}\cdot \frac{x^{\frac{1}{2}}+x\frac{1}{4}}{x^{\frac{1}{2}}+1}\cdot x^{\frac{1}{4}}+1=\frac{x^{\frac{1}{2}}-1}{x^{\frac{2}{4}}\left ( x^{\frac{1}{4}}+1 \right )}\cdot \frac{x^{\frac{1}{4}}\left ( x^{\frac{1}{4}}+1 \right )}{1}\cdot x^{\frac{1}{4}}+1=x^{\frac{1}{2}}-1+1=x^{\frac{1}{2}}=\sqrt{x}\)

Ответ: \(\sqrt{x}\)