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: \(1-x^{2}\)