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

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

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: \(-2a;2a\)

Решение №17108: \(\frac{a+b}{\left ( \sqrt{a}-\sqrt{b} \right )^{2}}\left ( \frac{3ab-b\sqrt{ab}+a\sqrt{ab}-3b^{2}}{\frac{1}{2}\sqrt{\frac{1}{4}}\left ( \frac{a}{b}+\frac{b}{a} \right )^{2}-1}+\frac{4ab\sqrt{a}+9ab\sqrt{b}-9b^{2}\sqrt{a}}{\frac{3}{2}\sqrt{b}-2\sqrt{a}} \right )=\frac{a+b}{\left ( \sqrt{a}-\sqrt{b} \right )^{2}}\left ( \frac{\left ( a-b \right )\left ( 3b+\sqrt{ab} \right )}{\frac{1}{4ab}\sqrt{\left ( a^{2}-b^{2} \right )^{2}}}-2\sqrt{ab}\left ( \sqrt{a}+3\sqrt{b} \right ) \right )=\frac{a+b}{\left ( \sqrt{a}-\sqrt{b} \right )^{2}}\left ( \frac{4ab\left ( a-b \right )\left ( 3b +\sqrt{ab}\right )}{a^{2}-b^{2}}-2\sqrt{ab}\left ( \sqrt{a}+3\sqrt{b} \right ) \right )=\frac{a+b}{a-2\sqrt{ab}+b}\cdot 2\sqrt{ab}\left ( \frac{2\sqrt{a}\left ( 3b+\sqrt{a} \right )}{a+b}-\left ( \sqrt{a}+3\sqrt{b} \right ) \right )=\frac{-2\sqrt{ab}\left ( a\sqrt{a}-5\sqrt{ab}+a\sqrt{b}+3b\sqrt{b} \right )}{a-2\sqrt{ab}+b}=-2\sqrt{ab}\left ( \sqrt{a}+3\sqrt{b} \right )=-2b\left ( a+3\sqrt{ab} \right )\)

Ответ: \(-2b\left ( a+3\sqrt{ab} \right )\)

Решение №17109: \(\frac{11-6\sqrt{2}}{\sqrt[3]{45-29\sqrt{2}}}=3-\sqrt{2};\frac{9-6\sqrt{2}+2}{\sqrt[3]{27-27\sqrt{2}+18-2\sqrt{2}}}=\frac{\left ( 3-\sqrt{2} \right )^{2}}{\left ( 3-\sqrt{2} \right )}=3-\sqrt{2};3-\sqrt{2}=3-\sqrt{2} \)

Ответ: \(3-\sqrt{2}=3-\sqrt{2}\)

Решение №17110: \(19=x^{3}-y^{3}=\left ( x-y \right )\left ( x^{2}+xy+y^{2} \right )=n*m; m=x^{2}+xy+y^{2};\left ( y+1 \right )^{2}+y\left ( y+1 \right )+y^{2}=19;y^{2}+y-6=0;3^{3}-2^{3}\)

Ответ: \(3^{3}-2^{3}\)

Решение №17111: \(\left ( \frac{x-9}{x+3x^{\frac{1}{2}}+9}:\frac{x^{0.5}+3}{x^{1.5}-27} \right )^{0.5}-x^{0.5}=\left ( \frac{x-9}{\sqrt{x}^{2}+3\sqrt{x}+9}:\frac{\sqrt{x}+3}{\sqrt{x}^{2}-3^{3}} \right )^{0.5}-\sqrt{x}=\sqrt{\left ( \sqrt{x}-3 \right )^{2}}-\sqrt{x}=\left | \sqrt{x}-3 \right |-\sqrt{x}=3-2\sqrt{x};-3\)

Ответ: \(3-2\sqrt{x};-3\)

Решение №17112: \(u-v=a;u^{2}-v^{2}=b;u^{3}-v^{3}=c;=u-v=a;\left ( u-v \right )\left ( u+v \right )=b;\left ( u-v \right )\left ( \left ( u+v \right )^{2}-uv \right )=c;3b^{2}+a^{4}=4ac\)

Ответ: \(3b^{2}+a^{4}=4a\)

Решение №17113: \(\left ( \sqrt[4]{mn^{2}p}+m\sqrt{\frac{3n}{m}}+\sqrt{3np} \right )\left ( \sqrt[4]{36mn^{2}p}-\sqrt{3mn}-p\sqrt{\frac{3n}{p}} \right )=\left ( \sqrt[4]{36mn^{2}p}+\left ( \sqrt{3mn}+\sqrt{3np} \right ) \right )\left ( \sqrt[4]{36mn^{2}p}-\left ( \sqrt{3mn}+\sqrt{3np} \right ) \right )=\left ( \sqrt[4]{36mn^{2}p} \right )^{2}-\left ( \sqrt{3mn}+\sqrt{3np} \right )^{2}=\sqrt{36mn^{2}p}-3mn-2\sqrt{9mn^{2}p}-3np=-3n\left ( m+p \right )\)

Ответ: \(-3n\left ( m+p \right )\)

Решение №17114: \(\left ( \frac{1}{\left ( x+3 \right )^{2}}\left ( \frac{1}{x^{2}}+\frac{1}{9} \right )+\frac{2}{\left ( x+3 \right )^{3}} \left ( \frac{1}{x}+\frac{1}{3} \right )\right )^{-\frac{1}{2}}=\left ( \frac{x^{2}+9}{9x^{2}\left ( x+3 \right )^{2}}+\frac{2}{3x\left ( x+3 \right )^{2}} \right )^{-\frac{1}{2}}=\left ( \frac{\left ( x+3 \right )^{2}}{9x^{2}\left ( x+3 \right )^{2}} \right )^{-\frac{1}{2}}=\left ( \frac{1}{9x^{2}} \right )^{-\frac{1}{2}}=\sqrt{9x^{2}}=3\left | x \right |=-3x;-3x\)

Ответ: \(-3x;-3x\)

Решение №17115: \(\left ( x^{2}-6x+1+\left ( \frac{\frac{x-3}{1+3x}-\frac{x-5}{1+5x}}{1+\frac{\left ( x-5 \right )\left ( x-3 \right )}{\left ( 1+5x \right )\left ( 1+3x \right )}} \right )^{-1} \right )^{\frac{1}{2}}=\left ( x^{2}-6x+1+\left ( \frac{\left ( x-3 \right )\left ( 1+5x \right )-\left ( x-5 \right )\left ( 1+3x \right )}{\left ( 1+5x \right )\left ( 1+3x \right )+\left ( x-5 \right )\left ( x-3 \right )} \right )^{-1} \right )^{\frac{1}{2}}=\left ( x^{2}-6x+1+\left ( \frac{2x^{2}+2}{16x^{2}+16} \right )^{-1} \right )^{\frac{1}{2}}=\left ( x^{2}-6x+1+\left ( \frac{1}{8} \right )^{-1} \right )^{\frac{1}{2}}=\left ( x^{2}-6x+1+8 \right )^{\frac{1}{2}}=\sqrt{\left ( x-3 \right )^{2}}=\left | x-3 \right |=3-x;x-3\)

Ответ: \(3-x;x-3\)

Решение №17116: \(\frac{1+\sqrt{1+x}}{x+1}+\frac{1+\sqrt{1-x}}{x-1}=\frac{x+x\sqrt{1+x}-1-\sqrt{x+1}+x+x\sqrt{1-x}+1+\sqrt{1-x}}{x^{2}-1}=\frac{2x-\left ( 1-x \right )\sqrt{1+x}+\left ( 1+x \right )\sqrt{1-x}}{x^{2}-1}=\frac{\sqrt{3}-\left ( 1-\frac{\sqrt{3}}{2} \right )\sqrt{1-\frac{\sqrt{3}}{2}}+\left ( 1+\frac{\sqrt{3}}{2}\sqrt{1-\frac{\sqrt{3}}{2}} \right )}{\frac{3}{4}-1}=-4\sqrt{3}+2-\sqrt{3}+2\sqrt{3}-3+2+\sqrt{3}-2\sqrt{3}-3=-4\sqrt{3}-2\)

Ответ: \(-4\sqrt{3}-2\)

Решение №17117: \(\frac{\sqrt{\left ( 2p+1 \right )^{3}}+\sqrt{\left ( 2p-1 \right )^{3}}}{\sqrt{4p+2\sqrt{4p^{2}-1}}}=\frac{\left ( \sqrt{2p+1}+\sqrt{2p-1} \right )\left ( \left ( \sqrt{2p+1} \right )^{2}-\sqrt{2p+1}\sqrt{2p-1}+\left ( \sqrt{2p-1} \right )^{2}\right ) }{\sqrt{2p+1+2\sqrt{4p^{2}-1}}+2p-1}=\frac{\left ( \sqrt{2p+1}+\sqrt{2p-1} \right )\left ( 4p-\sqrt{4p^{2}-1} \right )}{\sqrt{\left ( \sqrt{2p+1}+\sqrt{2p-1} \right )^{2}}}=4p-\sqrt{4p^{2}-1}\)

Ответ: \(4p-\sqrt{4p^{2}-1}\)

Решение №17118: \(\frac{2\sqrt[3]{2}}{1+\sqrt{3}}=\frac{\sqrt[3]{20+12\sqrt{3}}}{2+\sqrt{3}};\left (\frac{2\sqrt[3]{2}}{1+\sqrt{3}} \right )^{3}=\left (\frac{\sqrt[3]{20+12\sqrt{3}}}{2+\sqrt{3}} \right )^{3};\frac{16}{10+6\sqrt{3}}=\frac{20+12\sqrt{3}}{26+15\sqrt{3}};\frac{2}{5+3\sqrt{3}}=\frac{5+3\sqrt{3}}{26+15\sqrt{3}};2\left ( 26+15\sqrt{3} \right )=\left ( 5+3\sqrt{3} \right )\left ( 5+3\sqrt{3} \right );52+30\sqrt{3}=25+30\sqrt{3}+27;52+30\sqrt{3}=52+30\sqrt{3}\)

Ответ: \(52+30\sqrt{3}=52+30\sqrt{3}\)

Решение №17119: \(\left ( \frac{4}{3-\sqrt{5}} \right )^{2}-\left ( \frac{6-5\sqrt{6}}{5-\sqrt{6}} \right )^{2}=2\sqrt{61+24\sqrt{5}}; \frac{16}{14-6\sqrt{5}}-6=2\sqrt{61+24\sqrt{5}};\frac{8}{7-3\sqrt{5}}-6=2\sqrt{61+24\sqrt{5}}; \frac{4-21+9\sqrt{5}}{7-3\sqrt{5}}=\sqrt{61+24\sqrt{5}};\frac{12\sqrt{5}+16}{4}=\sqrt{61+24\sqrt{5}}; 3\sqrt{5}+4=\sqrt{61+24\sqrt{5}}; 45+24\sqrt{5}+16=61+24\sqrt{5};61+24\sqrt{5}=61+24\sqrt{5}\)

Ответ: \(61+24\sqrt{5}=61+24\sqrt{5}\)

Решение №17120: \(\frac{\sqrt{2}-1}{\sqrt{2}+1}=\sqrt[3]{\frac{10-7\sqrt{2}}{10+7\sqrt{2}}}; \frac{\left ( \sqrt{2}-1 \right )\left ( \sqrt{2}-1 \right )}{\left ( \sqrt{2}+1 \right )\left ( \sqrt{2}-1 \right )}=\sqrt[3]{\frac{\left ( 10-7\sqrt{2} \right )\left ( 10-7\sqrt{2} \right )}{\left ( 10+7\sqrt{2} \right )\left ( 10-7\sqrt{2} \right )}}; 2-2\sqrt{2}+1=\sqrt[3]{\frac{100-140\sqrt{2}+98}{2}};3-2\sqrt{2}=\sqrt[3]{99-70\sqrt{2}};27-54\sqrt{2}+72-12\sqrt{2}=99-70\sqrt{2};99-70\sqrt{2}=99-70\sqrt{2}\)

Ответ: \(99-70\sqrt{2}=99-70\sqrt{2}\)

Решение №17121: \(\frac{a^{\frac{7}{3}}-2a^{\frac{5}{3}}b^{\frac{2}{3}}+ab^{\frac{4}{3}}}{a^{\frac{5}{3}}-a^{\frac{4}{3}}b^{\frac{1}{3}}-ab^{\frac{2}{3}}+a^{\frac{2}{3}}b}:a^{\frac{1}{3}}=\frac{a^{\frac{3}{3}}\left (a^{\frac{4}{3}}-2a^{\frac{2}{3}}b^{\frac{2}{3}}+b^{\frac{4}{3}} \right )}{a^{\frac{2}{3}}\left ( a^{\frac{3}{3}}-a^{\frac{2}{3}}b^{\frac{1}{3}}-a^{\frac{1}{3}}b^{\frac{2}{3}}+b^{\frac{3}{3}} \right )}\cdot \frac{1}{a^{\frac{1}{3}}}=\frac{\left ( a^{\frac{2}{3}}-b^{\frac{2}{3}} \right )^{2}}{\left ( a^{\frac{3}{3}}-a^{\frac{2}{3}}b^{\frac{1}{3}} \right )-\left ( a^{\frac{1}{3}}b^{\frac{2}{3}}+b^{\frac{3}{3}} \right )}=\frac{\left (a^{\frac{2}{3}}-b^{\frac{2}{3}} \right )}{\left ( a^{\frac{1}{3}}-b^{\frac{1}{3}} \right )\left ( a^{\frac{2}{3}}-b^{\frac{2}{3}} \right )}=\frac{a^{\frac{2}{3}}-b^{\frac{2}{3}}}{a^{\frac{1}{3}}-b^{\frac{1}{3}}}=a^{\frac{1}{3}}+b^{\frac{1}{3}}\)

Ответ: \(a^{\frac{1}{3}}+b^{\frac{1}{3}}\)

Решение №17122: \(\frac{a^{\frac{1}{2}}+ab^{-1}}{a^{-\frac{1}{3}}-a^{-\frac{1}{6}}b^{-\frac{2}{3}}+b^{-\frac{2}{3}}}-\frac{a}{\sqrt[3]{b}}=\frac{\sqrt[6]{a^{5}}\left ( \sqrt[6]{a}+\sqrt[6]{b^{2}}-\sqrt[6]{a} \right )}{\sqrt[6]{b^{2}}}=\frac{\sqrt[6]{a^{5}}\sqrt[6]{b^{2}}}{\sqrt[6]{b^{2}}}=\sqrt[6]{a^{5}}=a^{\frac{5}{6}}\)

Ответ: \(a^{\frac{5}{6}}\)

Решение №17123: \(\left ( \left ( a^{\frac{1}{2}}-b^{\frac{1}{2}} \right )^{-1}\left ( a^{\frac{3}{2}}-b^{\frac{3}{2}} \right )-\frac{1}{\left ( \sqrt{a}+\sqrt{b} \right )^{-2}} \right ):\sqrt[3]{ab\sqrt{ab}}+\frac{1}{1+\left ( a\left ( 1-a^{2} \right )^{-\frac{1}{2}} \right )^{2}}=\left ( \frac{a^{\frac{3}{2}}-b^{\frac{3}{2}}}{a^{\frac{1}{2}}-b^{\frac{1}{2}}}-\left ( \sqrt{a}+\sqrt{b} \right )^{2} \right ):\sqrt{ab}+\frac{1}{1+\frac{a^{2}}{1-a^{2}}}=\left ( a+a^{\frac{1}{2}}b^{\frac{1}{2}}+b-a-2a^{\frac{1}{2}}b^{\frac{1}{2}}-b \right )\cdot \frac{1}{a^{\frac{1}{2}}b^{\frac{1}{2}}}+1-a^{2}=-a^{\frac{1}{2}}b^{\frac{1}{2}}\cdot \frac{1}{a^{\frac{1}{2}}b^{\frac{1}{2}}}+1-a^{2}=1+1-a^{2}=-a^{2}\)

Ответ: \(-a^{2}\)

Решение №17124: \(\frac{\left ( ab\left ( x^{2}+y^{2} \right )+xy\left ( a^{2}+b^{2} \right ) \right )\left ( \left ( ax+by \right )^{2}-4abxy \right )}{ab\left ( x^{2}-y^{2} \right )+xy\left ( a^{2}-b^{2} \right )}=\frac{-\left ( a^{2}-b^{2} \right )y\frac{+}{}y\sqrt{a^{4}-2a^{2}b^{2}+b^{4}+4a^{2}b^{2}}}{2ab}=\frac{\left ( -a^{2}+b^{2}\frac{+}{}\left ( a^{2}+b^{2} \right ) \right )y}{2ab}=\frac{2b^{2}y}{2ab};-\frac{2a^{2}y}{2ab}=-\frac{ay}{b};\frac{by}{a}=abx^{2}+\left ( a^{2}+b^{2} \right )yx+aby^{2}=ab\left ( x+\frac{ay}{b} \right )\left ( x-\frac{by}{a} \right )=\left ( abx+a^{2}y \right )\left ( abx-b^{2}y \right )=a^{2}x^{2}-b^{2}y^{2}\)

Ответ: \(a^{2}x^{2}-b^{2}y^{2}\)

Решение №17125: \(\frac{2a\sqrt{1+x^{2}}}{x+\sqrt{1+x^{2}}}; x=\frac{1}{2}\left ( \sqrt{\frac{a}{b}-\sqrt{\frac{b}{a}}} \right );=\frac{2a\sqrt{1+\left (\frac{1}{2}\left ( \sqrt{\frac{a}{b}-\sqrt{\frac{b}{a}}} \right )^{2}}}{\frac{1}{2}\left ( \sqrt{\frac{a}{b}-\sqrt{\frac{b}{a}}}+\sqrt{1+\left ( \frac{1}{2}\left ( \sqrt{\frac{a}{b}+\sqrt{\frac{b}{a}}} \right )^{2}}}=\frac{2a\sqrt{1+\frac{a^{2}-2ab+b^{2}}{4ab}}}{\frac{a-b}{2\sqrt{ab}}+\sqrt{1+\frac{a^{2}+2ab+b^{2}}{4ab}}}=\frac{a\left ( a+b \right )}{\sqrt{ab}}:\left ( \frac{a-b+a+b}{2\sqrt{ab}} \right )=\frac{a\left ( a+b \right )}{\sqrt{ab}}:\frac{2a}{2\sqrt{ab}}=\frac{a\left ( a+b \right )}{\sqrt{ab}}\cdot \frac{\sqrt{ab}}{a}=a+b\)

Ответ: \(a+b\)

Решение №17126: \(16x^{2}+144x+\left ( a+b \right );b-a=-7;144^{2}-4\cdot 16\left ( a+b \right )=0;b-a=-7;b+a=324;a=165.5;b=158.5\)

Ответ: \(a=165.5;b=158.5\)

Решение №17127: \(\frac{\left ( a^{2}-b^{2} \right )\left ( \sqrt[3]{a}-\sqrt[3]{b} \right )}{\sqrt[3]{a^{4}}+\sqrt[3]{ab^{3}}-\sqrt[3]{a^{3}b}-\sqrt[3]{b^{4}}}=\frac{\left ( a+b \right )\left ( a-b \right )\left ( \sqrt[3]{a}-\sqrt[3]{b} \right )}{\left ( a+b \right )\left ( \sqrt[3]{a}-\sqrt[3]{b} \right )}=a-b\)

Ответ: \(a-b\)

Решение №17128: \(\frac{2b\sqrt{x^{2}-1}}{x-\sqrt{x^{2}-1}}; x=\frac{1}{2}\left ( \sqrt{\frac{a}{b}+\sqrt{\frac{b}{a}}} \right );=\frac{2b\sqrt{\left (\frac{1}{2}\left ( \sqrt{\frac{a}{b}+\sqrt{\frac{b}{a}}} \right )^{2}-1}}{\frac{1}{2}\left ( \sqrt{\frac{a}{b}+\sqrt{\frac{b}{a}}}-\sqrt{\left ( \frac{1}{2}\left ( \sqrt{\frac{a}{b}+\sqrt{\frac{b}{a}}} \right )^{2}-1}}=\frac{2b\sqrt{\frac{\left ( a-b \right )^{2}}{4ab}}}{\frac{a+b}{2\sqrt{ab}-\sqrt{\frac{\left ( a-b \right )^{2}}{4ab}}}}=\frac{b\left ( a-b \right )}{\sqrt{ab}}\cdot \frac{2\sqrt{ab}}{2b}=a-b\)

Ответ: \(a-b\)

Решение №17129: \(\frac{a\left ( a-2 \right )-b\left ( b+2 \right )+\sqrt{ab}\left ( b-a+2 \right )}{a+b-\sqrt{ab}}:\left ( 1+2\frac{a^{2}+b^{2}+ab}{b^{3}-a^{3}} \right )=\frac{a^{2}-2a-b^{2}-2b-\sqrt{ab}\left ( a-b-2 \right )}{a+b-\sqrt{ab}}:\left ( 1-2\frac{a^{2}+b^{2}+ab}{b^{3}-a^{3}} \right )=\frac{\left ( a-b \right )\left ( a+b \right )-2\left ( a+b \right )-\sqrt{ab}\left ( a-b-2 \right )}{a+b-\sqrt{ab}}:\left ( 1-\frac{2}{a-b} \right )=\frac{\left ( a-b-2 \right )\left ( a+b-\sqrt{ab} \right )}{a+b-\sqrt{ab}}\cdot \frac{a-b}{a-b-2}=a-b\)

Ответ: \(a-b\)

Решение №17130: \(\left ( \sqrt[3]{m^{2}}+n\sqrt[3]{m}+n^{2} \right )\cdot \frac{\sqrt[3]{m^{4}}-n^{3}+n^{2}\sqrt[3]{m}-mn}{mn^{-1}+n-n^{4}m^{-1}-n^{2}}=\left ( m^{\frac{2}{3}}+nm^{\frac{1}{3}}+n^{2} \right )\frac{m^{\frac{4}{3}}-n^{3}+n^{2}m^{\frac{1}{3}}-mn}{\frac{m}{n}+n-\frac{n^{4}}{m}-n^{2}}=\left ( m^{\frac{2}{3}}+nm^{\frac{1}{3}}+n^{2} \right )\frac{\left ( m+n^{2} \right )\left ( m^{\frac{1}{3}}-n \right )}{\frac{\left ( m+n^{2} \right )\left ( m-n^{3} \right )}{mn}}=\frac{\left ( m-n^{3} \right )mn}{m-n^{3}}=mn\)

Ответ: \(mn\)

Решение №17131: \(\left ( \frac{pq^{2}}{\left ( p+q \right )^{\frac{5}{2}}}-\frac{2pq^{2}}{\left ( p+q \right )^{\frac{3}{2}}}+\frac{pq}{\sqrt{p+q}} \right ):\left ( \frac{p^{2}}{\left ( p+q \right )^{\frac{5}{2}}}-\frac{p^{2}q}{\left ( p+q \right )^{\frac{7}{2}}} \right )=\frac{pq}{\left ( p+q \right )^{\frac{1}{2}}}\cdot \left ( \frac{q^{2}}{\left ( p+q \right )^{2}} -\frac{2q}{p+q}+1\right ):\frac{p^{2}}{\left ( p+q \right )^{\frac{5}{2}}}\cdot \left ( 1-\frac{q}{p+q} \right )=\frac{pq}{\left ( p+q \right )^{\frac{1}{2}}}\left ( \frac{q^{2}-2q\left ( p+q \right )+\left ( p+q \right )^{2}}{\left ( p+q \right )^{2}} \right ):\frac{p^{2}}{\left ( p+q \right )^{\frac{5}{2}}}\left ( \frac{p+q-q}{p+q} \right )=\frac{p^{3}q}{\left ( p+q \right )^{\frac{5}{2}}}\cdot \frac{\left ( p+q \right )^{\frac{5}{2}}\left ( p+q \right )}{p^{3}}=q\left ( p+q \right )\)

Ответ: \(q\left ( p+q \right )\)

Решение №17132: \(\frac{\sqrt{1+\sqrt{1-x^{2}}}\left ( \sqrt{\left ( 1+x \right )^{3}}-\sqrt{\left ( 1-x \right )^{3}} \right )}{2+\sqrt{1-x^{2}}}=\sqrt{1+\sqrt{1-x^{2}}}\left ( \sqrt{1+x}-\sqrt{1-x} \right )=\sqrt{\frac{1+x+2\sqrt{1-x^{2}+1-x}}{2}}\left ( \sqrt{1+x}-\sqrt{1-x} \right )=\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{2}}\left ( \sqrt{1+x}-\sqrt{1-x} \right )=\frac{1+x-1+x}{\sqrt{2}}=\frac{2x}{\sqrt{2}}=x\sqrt{2}\)

Ответ: \(x\sqrt{2}\)

Решение №17133: \(x\sqrt[3]{2x\sqrt{xy}-x\sqrt{3xy}}\cdot \sqrt[6]{x^{2}y\left ( 7+4\sqrt{3} \right )}=x\sqrt[3]{x\sqrt{xy\left ( 2-\sqrt{3} \right )}}\cdot \sqrt[6]{x^{3}y\left ( 7+4\sqrt{3} \right )}=x\sqrt[3]{x\sqrt{xy}\left ( 2-\sqrt{3} \right )}\cdot \sqrt[6]{\left | x \right |\sqrt{xy}\left ( 2+\sqrt{3} \right )}=x\sqrt[3]{x\left | x \right |\left | x \right |\left | y \right |}=x\sqrt[3]{x^{3}\left | y \right |}=x\cdot x\cdot \left | \sqrt[3]{y} \right |=x^{2}\left | \sqrt[3]{y} \right |\)

Ответ: \(x^{2}\left | \sqrt[3]{y} \right |\)

Решение №17134: \(\frac{x^{4}+x^{2}+x\sqrt{2}+2}{x^{2}-x\sqrt{2}+2}-x\sqrt{2}=\frac{x^{4}+x^{2}+x\sqrt{2}+2-x^{3}\sqrt{2}+2x^{2}-2x\sqrt{2}}{x^{2}-x\sqrt{2}+2}=\frac{x^{4}-\sqrt{2}x^{3}+3x^{2}-\sqrt{2}x+2}{x^{2}-x\sqrt{2}+2}=\frac{\left ( x^{2}-\sqrt{2}x+2 \right )\left ( x^{2}+1 \right )}{x^{2}-x\sqrt{2}+2}=x^{2}+1\)

Ответ: \(x^{2}+1\)

Решение №17135: \(\frac{\left ( \sqrt{a}+\sqrt{ax}+x+x\sqrt{x} \right )^{2}\left ( 1-\sqrt{x} \right )^{2}}{\left ( x+x^{-1}-2 \right )a^{-\frac{1}{4}}}-\frac{\left ( x\sqrt{a} \right )^{\frac{3}{2}}}{\left ( ax^{-1}+4\sqrt{a}+4x \right )^{-\frac{1}{2}}}=\frac{\left ( \sqrt{a}\left ( 1+\sqrt{x} \right ) +x\left ( 1+\sqrt{x} \right )\right )^{2}\left ( 1-\sqrt{x} \right )^{2}}{\left ( x+\frac{1}{x}-2 \right )\frac{1}{\sqrt[4]{a}}}-\frac{\sqrt{x^{3}}\sqrt[4]{a^{3}}}{\left ( \frac{a}{x}+4\sqrt{a}+4x \right )^{-\frac{1}{2}}}=\left ( \sqrt{a}+x \right )^{2}\cdot x\sqrt[4]{a}-x\sqrt[4]{a^{3}}\left ( \sqrt{a}+2x \right )=x\sqrt[4]{a}\left ( \left ( \sqrt{a}+x \right )^{2}-\sqrt[4]{a^{2}}\left ( \sqrt{a}+2x \right ) \right )=x\sqrt[4]{a}\left ( a+2\sqrt{a}\cdot x+x^{2}-a-2\sqrt{a}\cdot x \right )=x\sqrt[4]{a}\cdot x^{2}=x^{3}\cdot \sqrt[4]{a}\)

Ответ: \(x^{3}\cdot \sqrt[4]{a}\)

Решение №17136: \(\frac{x-1}{x+x^{\frac{1}{2}}+1}:\frac{x^{0.5}+1}{x^{1.5}-1}+\frac{2}{x^{-0.5}}=\frac{\left ( x^{\frac{1}{2}}-1 \right )\left ( x^{\frac{1}{2}}+1 \right )}{x+x^{\frac{1}{2}}+1}\cdot \frac{\left ( x^{\frac{1}{2}} \right )^{3}-1}{x^{\frac{1}{2}}+1}+\frac{2}{\frac{1}{x^{\frac{1}{2}}}}=\frac{\left ( x^{\frac{1}{2}}-1 \right )\left ( x^{\frac{1}{2}}+1 \right )\left (x+x^{\frac{1}{2}}+1 \right )}{x+x^{\frac{1}{2}}+1}+2x^{\frac{1}{2}}=\left ( x^{\frac{1}{2}}-1 \right )^{2}+2x^{\frac{1}{2}}=x-2x\frac{1}{2}+1+2x\frac{1}{2}=x+1\)

Ответ: \(x+1\)

Решение №17137: \(\frac{\sqrt[4]{x^{5}}+\sqrt[4]{xy^{4}}-\sqrt[4]{x^{4}y}-\sqrt[4]{x^{5}}}{\sqrt{x}-\sqrt{y}}\cdot \left ( \sqrt[4]{x}+\sqrt[4]{y} \right )=\frac{\sqrt[4]{x}\left ( \sqrt[4]{x^{4}}+\sqrt[4]{y^{4}} \right )-\sqrt[4]{y}\left ( \sqrt[4]{x^{4}}+\sqrt[4]{y^{4}} \right )}{\left ( \sqrt[4]{x}-\sqrt[4]{y} \right )\left ( \sqrt[4]{x}+\sqrt[4]{y} \right )}\cdot \left ( \sqrt[4]{x}+\sqrt[4]{y} \right )=\sqrt[4]{x^{4}}+\sqrt[4]{y^{4}}=x+y\)

Ответ: \(x+y\)

Решение №17138: \(\left ( \left ( \frac{2^{\frac{3}{2}}+27y^{\frac{3}{5}}}{\sqrt{2}+3\sqrt[5]{y}}+3\sqrt[10]{32y^{2}}-2 \right )\cdot 3^{-2} \right )^{5}=\left ( \left ( \frac{\left ( \sqrt{2} \right )^{3}+\left ( 3\sqrt[5]{y} \right )^{3}}{\sqrt{2}+3\sqrt[5]{y}}+3\sqrt{2}\sqrt[5]{y}-2 \right )\cdot \frac{1}{9} \right )^{5}=\left ( \left ( \frac{\left ( \sqrt{2}+3\sqrt[5]{y} \right )\left ( \left ( \sqrt{2} \right )^{2}-3\sqrt{2}\sqrt[5]{y}+\left ( 3\sqrt[5]{y} \right )^{2} \right )}{\sqrt{2}+3\sqrt[5]{y}}+3\sqrt{2}\sqrt[5]{y}-2 \right ) \cdot \frac{1}{9}\right )^{5}=\left ( 9\sqrt[5]{y} \cdot \frac{1}{9}\right )^{5}=\left ( \sqrt[5]{y} \right )^{5}=y^{2}\)

Ответ: \(y^{2}\)