Решение №16957: \(\frac{2\left | a+5 \right |-a+\frac{25}{a}}{3a^{2}+10a-25}=\frac{2a\left | a+5 \right |-a^{2}+25}{a\left ( 3a^{2}+10a-25 \right )}=\frac{3\left ( 3a^{2}+10a-25 \right )}{a\left ( 3a^{2}+10a-25 \right )};\frac{a^{2}+10a+25}{a\left ( a+5 \right )\left ( 3a-5 \right )}=-\frac{1}{a};\frac{a+5}{a\left ( 3a-5 \right )}\)

Ответ: \(-\frac{1}{a};\frac{a+5}{a\left ( 3a-5 \right )}\)

Решение №16958: \(\left ( \frac{\sqrt[4]{a^{3}}-1}{\sqrt[4]{a}-1}+\sqrt[4]{a} \right )^{\frac{1}{2}}\cdot \left ( \frac{\sqrt[4]{a^{3}}+1}{\sqrt[4]{a}+1}-\sqrt{a} \right )\cdot \left ( a-\sqrt{a^{3}} \right )^{-1}=\left ( \frac{\left ( \sqrt[4]{a}-1 \right )\left ( \sqrt[4]{a^{2}}+\sqrt[4]{a}+1 \right )}{\sqrt[4]{a}-1}+\sqrt[4]{a} \right )^{\frac{1}{2}}\cdot \left ( \frac{\left ( \sqrt[4]{a}+1 \right )\left ( \sqrt[4]{a^{2}}-\sqrt[4]{a}+1 \right )}{\sqrt[4]{a}+1}-\sqrt{a} \right )\cdot \frac{1}{a-\sqrt{a^{3}}}=\left ( \sqrt[4]{a^{2}}+\sqrt[4]{a}+1+\sqrt[4]{a} \right )^{\frac{1}{2}}\cdot \left ( \sqrt[4]{a^{2}}-\sqrt[4]{a}+1-\sqrt[4]{a^{2}} \right )\cdot \frac{1}{a-\sqrt{a^{3}}}=\frac{\left ( \sqrt[4]{a}+1 \right )\left ( 1-\sqrt[4]{a} \right )}{a\left ( 1-\sqrt[4]{a} \right )\left ( 1+\sqrt[4]{a} \right )}=\frac{1}{a}\)

Ответ: \(\frac{1}{a}\)

Решение №16959: \(\frac{a^{2}-4-\left | a-2 \right |}{a^{3}+2a^{2}-5a-6}=\frac{\left ( a-2 \right )\left ( a+2 \right )-\left | a-2 \right |}{\left ( a-2 \right )\left ( a+3 \right )\left ( a+1 \right )}=\frac{a+3}{\left ( a+3 \right )\left ( a+1 \right )};\frac{a+1}{\left ( a+3 \right )\left ( a+1 \right )}=\frac{1}{a+1};\frac{1}{a+3}\)

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

Решение №16960: \(\frac{\left ( \sqrt{a}+\sqrt{b} \right )^{2}-4b}{\left ( a-b \right ):\left ( \sqrt{\frac{1}{b}}+3\sqrt{\frac{1}{a}} \right )}:\frac{a+9b+6\sqrt{ab}}{\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{a}}}=\frac{\frac{a+2\sqrt{ab}+b-4b}{\left ( \sqrt{a}-\sqrt{b} \right )\left ( \sqrt{a}+\sqrt{b} \right ):\left ( \frac{1}{\sqrt{b}}+\frac{3}{\sqrt{a}} \right )}}{\frac{\left ( \sqrt{a}+3\sqrt{b} \right )^{2}}{\frac{\sqrt{a}+\sqrt{b}}{\sqrt{ab}}}}=\frac{a+2\sqrt{ab}-3b}{\left ( \sqrt{a}-\sqrt{b} \right )\left ( \sqrt{a}+\sqrt{b} \right ):\frac{\sqrt{a}+3\sqrt{b}}{\sqrt{ab}}}:\frac{\left ( \sqrt{a}+3\sqrt{b} \right )^{2}\sqrt{ab}}{\left ( \sqrt{a}+\sqrt{b} \right )}=\frac{a+2\sqrt{ab}-3b}{ab\left ( a-\sqrt{ab}+3\sqrt{ab-3b} \right )}=\frac{1}{ab}\)

Ответ: \(\frac{1}{ab}\)

Решение №16961: \(\left ( \left ( \frac{a\sqrt[3]{b}}{b\sqrt{a^{3}}} \right )^{\frac{3}{2}}+\left ( \frac{\sqrt{a}}{a\sqrt[8]{b^{3}}} \right )^{2} \right ):\left ( a^{\frac{1}{4}}+b^{\frac{1}{4}} \right )=\left ( \frac{a^{\frac{3}{2}}\left ( \sqrt[3]{b} \right )^{\frac{3}{2}}}{b^{\frac{3}{2}}\left ( \sqrt{a^{3}} \right )^{\frac{3}{2}}}+\frac{\left ( \sqrt{a} \right )^{2}}{a^{2}\left ( \sqrt[8]{b^{3}} \right )^{2}} \right ):\left ( a^{\frac{1}{4}}+b^{\frac{1}{4}} \right )=\left ( \frac{a^{\frac{3}{2}}b^{\frac{1}{2}}}{a^{\frac{9}{4}}b^{\frac{3}{2}}}+\frac{a}{a^{2}b^{\frac{3}{4}}} \right )\frac{1}{a^{\frac{1}{4}}+b^{\frac{1}{4}}}=\left ( \frac{1}{a^{\frac{3}{4}}b}+\frac{1}{ab^{\frac{3}{4}}} \right )\frac{1}{a^{\frac{1}{4}}+b^{\frac{1}{4}}}=\frac{1}{ab}\)

Ответ: \(\frac{1}{ab}\)

Решение №16962: \(\left ( \left ( \sqrt{mn}-\frac{mn}{m+\sqrt{mn}} \right ):\frac{\sqrt[4]{mn}-\sqrt{n}}{m-n}-m\sqrt{n} \right )^{2}:\sqrt[3]{mn\sqrt{mn}}-\left ( \frac{m}{\sqrt{m^{4}-1}} \right )^{-2}=\left ( \sqrt{mn}\left ( 1-\frac{\sqrt{mn}}{\left ( \sqrt{m}+\sqrt{} \right )\sqrt{m}} \right ):\frac{\sqrt[4]{n}\left ( \sqrt[4]{m}-\sqrt[4]{n} \right )}{\left ( \sqrt{m}-\sqrt{n} \right )\left ( \sqrt{m}+\sqrt{n} \right )} -m\sqrt{n}\right )^{2}:\sqrt{mn}-\frac{m^{4}-1}{m^{2}}=\left ( m\sqrt[4]{n}\left ( \sqrt[4]{m}+\sqrt[4]{n} \right )-m\sqrt{n} \right )^{2}\cdot \frac{1}{\sqrt{mn}}-\frac{m^{4}-1}{m^{2}}=m^{2}-\frac{m^{4}-1}{m^{2}}=\frac{m^{4}-m^{4}+1}{m^{2}}=\frac{1}{m^{2}}\)

Ответ: \(\frac{1}{m^{2}}\)

Решение №16963: \(\frac{m\left | m-3 \right |}{\left ( m^{2}-m-6 \right )\left | m \right |}; 1)\frac{m\left | m-3 \right |}{\left ( m^{2}-m-6 \right )\left | m \right |}=\frac{-m\left ( m-3 \right )}{-\left ( m^{2}-m-6 \right )m}=\frac{m-3}{\left ( m-3 \right )\left ( m+2 \right )}=\frac{1}{m+2} 2)\frac{m\left | m-3 \right |}{\left ( m^{2}-m-6 \right )\left | m \right |}=\frac{-m\left ( m-3 \right )}{\left ( m^{2}-m-6 \right )m}=\frac{-m+3}{\left ( m-3 \right )\left ( m+2 \right )}=-\frac{1}{m+2} 3)\frac{m\left | m-3 \right |}{\left ( m^{2}-m-6 \right )\left | m \right |}=\frac{-m\left ( m-3 \right )}{-\left ( m^{2}-m-6 \right )m}=\frac{1}{m+2} \frac{1}{m+2};-\frac{1}{m+2}\)

Ответ: \(\frac{1}{m+2};-\frac{1}{m+2}\)

Решение №16964: \(\frac{\sqrt{4x+4+x^{-1}}}{\sqrt{x}\left | 2x^{2}-x-1 \right |}=\frac{\sqrt{4x+4+\frac{1}{x}}}{\sqrt{x}\left | \left ( x-1 \right )\left ( 2x+1 \right ) \right |}=\frac{\sqrt{\left ( 2x+1 \right )^{2}}}{x\left | \left ( x-1 \right )\left ( 2x+1 \right ) \right |}=\frac{2x+1}{x\left ( 2x+1 \right )\left | x-1 \right |}=\frac{1}{x\left | x-1 \right |}=-\frac{1}{x\left ( x-1 \right )};\frac{1}{x\left ( x-1 \right )}\)

Ответ: \(-\frac{1}{x\left ( x-1 \right )};\frac{1}{x\left ( x-1 \right )}\)

Решение №16965: \(\frac{\left ( \sqrt{x}+\sqrt{2} \right )^{2}-\sqrt{2x}}{x^{2}+x-\sqrt{2x}+2}=\frac{x^{2}+2\sqrt{2x}+2-\sqrt{2x}}{\left ( x^{2}+x\sqrt{2x}+2x \right )-\left ( x\sqrt{2x}+\sqrt{2x}\sqrt{2x}+2\sqrt{2x} \right )+\left ( x+\sqrt{2x}+2 \right )}=\frac{x+\sqrt{2x}+2}{x\left ( x+\sqrt{2x}+2 \right )-\sqrt{2x}\left ( x+\sqrt{2x}+2 \right )+\left ( x+\sqrt{2x}+2 \right )}=\frac{x+\sqrt{2x}+2}{\left ( x+\sqrt{2x}+2 \right )\left ( x-\sqrt{2x}+1 \right )}=\frac{1}{x-\sqrt{2x}+1}\)

Ответ: \(\frac{1}{x-\sqrt{2x}+1}\)

Решение №16966: \(\left ( x\sqrt[3]{\frac{x-1}{\left ( x+1 \right )^{2}}}+\frac{x-1}{\sqrt[3]{\left ( x^{2}-1 \right )^{2}}} \right )^{-\frac{3}{5}}:\left ( x^{2}-1 \right )^{\frac{4}{5}}=\left ( \frac{x\sqrt[3]{x-1}}{\sqrt[3]{\left ( x+1 \right )^{2}}}+\frac{\sqrt[3]{x-1}}{\sqrt[3]{\left ( x+1 \right )^{2}}} \right )^{-\frac{3}{5}}\cdot \frac{1}{\sqrt[5]{\left ( x+1 \right )^{4}}}=\left ( \frac{\sqrt[3]{x-1}\left ( x+1 \right )}{\sqrt[3]{\left ( x+1 \right )^{2}}} \right )^{-\frac{3}{5}}\cdot \frac{1}{\sqrt[5]{\left ( x^{2}-1 \right )^{4}}}=\left ( \sqrt[3]{\left ( x-1 \right )\left ( x+1 \right )} \right )^{-\frac{3}{5}}\cdot \frac{1}{\sqrt[5]{\left ( x^{2}-1 \right )^{4}}}=\frac{1}{\sqrt[5]{\left ( x^{2}-1 \right )^{5}}}=\frac{1}{x^{2}-1}\)

Ответ: \(\frac{1}{x^{2}-1}\)