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 №17048: \(\frac{\left ( \sqrt{p^{3}}:\sqrt{p}+p \right )^{\frac{1}{4}}:\sqrt[8]{\left ( p-q \right )^{3}}}{\left ( \frac{\sqrt{q}}{\sqrt{p}-\sqrt{q}}-\sqrt{\frac{q}{p}}+1 \right )^{\frac{1}{4}}}=\frac{\sqrt[4]{\frac{\sqrt{p^{3}}}{\sqrt{p}}+p}:\sqrt[8]{\left ( p-q \right )^{3}}}{\sqrt[4]{\frac{\sqrt{q}}{\sqrt{p}-\sqrt{q}}-\frac{\sqrt{q}}{\sqrt{p}}+1}}=\sqrt[8]{\frac{\left ( \left ( \sqrt{p}+\sqrt{q} \right )\left ( p-\sqrt{pq}+q \right ) \right )^{2}\left ( \sqrt{p}-\sqrt{q} \right )^{2}}{\left ( p-q \right )^{3}\left ( p-\sqrt{pq}+q \right )^{2}}}=\sqrt[8]{\frac{\left ( \sqrt{p}+\sqrt{q} \right )^{2}\left ( \sqrt{p}-\sqrt{q} \right )^{2}}{\left ( p-q \right )^{3}}}=\sqrt[8]{\frac{\left ( p-q \right )^{2}}{\left ( p-q \right )^{3}}}=\sqrt[8]{\frac{1}{p-q}}-\frac{1}{\sqrt[8]{p-q}}\)

Ответ: \(\sqrt[8]{\frac{1}{p-q}}-\frac{1}{\sqrt[8]{p-q}}\)

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 №17050: \(\frac{x^{\frac{3}{p}}-x^{\frac{3}{q}}}{\left ( x^{\frac{1}{p}}+x^{\frac{1}{q}} \right )^{2}-2x^{\frac{1}{q}}\left ( x^{\frac{1}{p}}+x^{\frac{1}{q}} \right )}+\frac{x^{\frac{2}{p}}}{x^{\frac{q-p}{pq}}+1}=\frac{\left ( x^{\frac{1}{p}}-x^{\frac{1}{q}} \right )\left ( x^{\frac{2}{p}}+x^{\frac{1}{p}}x^{\frac{1}{q}}+x^{\frac{2}{q}} \right )}{\left ( x^{\frac{1}{p}}+x^{\frac{1}{q}} \right )\left ( x^{\frac{1}{p}}-x^{\frac{1}{q}} \right )}+\frac{x^{\frac{1}{p}}}{\frac{x^{\frac{1}{p}}}{x^{\frac{1}{q}}}+1}=x^{\frac{1}{p}}+x^{\frac{1}{q}}=\sqrt[p]{x}+\sqrt[q]{x}\)

Ответ: \(\sqrt[p]{x}+\sqrt[q]{x}\)

Решение №17051: \(\frac{2a\left ( a+2b+\sqrt{a^{2}+4ab} \right )}{\left ( a+\sqrt{a^{2}+4ab} \right )\left ( a+4b+\sqrt{a^{2}+4ab} \right )}=\frac{a+\sqrt{a^{2}+4ab}}{a+4b+\sqrt{a^{2}+4ab}}=\frac{a^{2}+4ab-a\sqrt{a^{2}+4ab}+a\sqrt{a^{2}+4ab}+4b\sqrt{a^{2}+4ab}-a^{2}-4ab}{\left ( a+4b \right )^{2}-\left ( \sqrt{a^{2}}+4ab \right )^{2}}=\frac{4b\sqrt{a^{2}+4ab}}{a^{2}+8ab+16b^{2}-a^{2}+4ab}=\frac{4b\sqrt{a^{2}+4ab}}{4ab+16b^{2}}=\frac{4b\sqrt{a^{2}+4ab}}{4b\left ( a+4b \right )}=\frac{\sqrt{a\left ( a+4b \right )}}{a+4b}=\sqrt{\frac{a\left ( a+4b \right )}{\left ( a+4b \right )^{2}}}=\sqrt{\frac{a}{a+4b}}\)

Ответ: \(\sqrt{\frac{a}{a+4b}}\)

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 №17071: \(\frac{\sqrt{\left ( 3x+2 \right )^{2}-24x}}{3\sqrt{x}-\frac{2}{\sqrt{x}}}=\frac{\sqrt{9x^{2}+12x+4-24x}}{\frac{3x-2}{\sqrt{x}}}=\frac{\sqrt{9x^{2}-12x+4}\sqrt{x}}{3x-2}=\frac{\sqrt{\left ( 3x-2 \right )^{2}}\sqrt{x}}{3x-2}=\frac{\left | 3x-2 \right |\sqrt{x}}{3x-2}=-\sqrt{x};\sqrt{x}\)

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: \(\sqrt{x}\)

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 №17086: \(\left ( \frac{9-4a^{-2}}{3a^{-\frac{1}{2}}}+2a^{-\frac{3}{2}}-\frac{1+a^{-1}-6a^{-2}}{a^{-\frac{1}{2}}+3a^{-\frac{3}{2}}} \right )^{4}=\left ( \frac{9a^{2}-4}{a^{2}}\cdot \frac{a^{\frac{3}{2}}}{3a+2}-\frac{a^{2}+a-6}{a^{2}}\cdot \frac{a^{\frac{3}{2}}}{a+3} \right )^{4}=\left ( \frac{3a-2}{a^{\frac{1}{2}}}-\frac{a-2}{a^{\frac{1}{2}}} \right )^{4}=\left ( \frac{2a}{a^{\frac{1}{2}}} \right )^{4}=\left ( 2a^{\frac{1}{2}} \right )^{4}=16a^{2}\)

Ответ: \(16a^{2}\)

Решение №17087: \(\left ( \frac{\left ( a-1 \right )^{-1}}{a^{-3}}-\left ( 1-a \right )^{-1} \right )\cdot \frac{1+a\left ( a-2 \right )}{a^{2}-a+1}\cdot \sqrt{\frac{1}{\left ( a+1 \right )^{2}}}=\left ( \frac{\frac{1}{a-1}}{\frac{1}{a^{3}}}-\frac{1}{1-a} \right )\cdot \frac{1+a^{2}-2a}{a^{2}-a+1}\cdot \frac{1}{\left | a+1 \right |}=\left ( \frac{a^{3}}{a-1}+\frac{1}{a-1} \right )\cdot \frac{a^{2}-2a+1}{a^{2}-a+1}\cdot \frac{1}{\left | a+1 \right |}=\frac{a^{3}+1}{a-1}\cdot \frac{\left ( a-1 \right )^{2}}{a^{2}-a+1}\cdot \frac{1}{\left | a+1 \right |}=\frac{\left ( a+1 \right )\left ( a-1 \right )}{\left | a+1 \right |}=1-a,a-1\)

Ответ: \(1-a,a-1\)

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: \(2;2\sqrt{x-1}\)