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: \(a-b\)