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

Решение №16967: \(\frac{x^{3}-6x^{2}+11x-6}{\left ( x^{3}-4x^{2}+3x \right )\left | x-2 \right |}=\frac{\left ( x-3 \right )\left ( x-2 \right )\left ( x-1 \right )}{x\left ( x-3 \right )\left ( x-1 \right )\left | x-2 \right |}=\frac{x-2}{x\left | x-2 \right |}=-\frac{1}{x};\frac{1}{x}\)

Ответ: \(-\frac{1}{x};\frac{1}{x}\)

Решение №16968: \(\left ( \frac{z-2}{6z+\left ( z-2 \right )^{2}}+\frac{\left ( z+4 \right )^{2}-12}{z^{3}-8}-\frac{1}{z-2} \right ):\frac{z^{3}+2z^{2}+2z+4}{z^{3}-2z^{2}+2z-4}=\left ( \frac{z-2}{6z+z^{2}-4z+4}+\frac{z^{2}+8z+16-12}{\left ( z-2 \right )\left ( z^{2}+2z+4 \right )}-\frac{2}{z-2} \right ):\frac{\left ( z+2 \right )\left ( z^{2}+2 \right )}{\left ( z-2 \right )\left ( z^{2}+2 \right )}=\frac{z^{2}+2z+4}{\left ( z+2 \right )\left ( z^{2}+2 \right )}\cdot \frac{z-2}{z+2}=\frac{1}{z+2}\)

Ответ: \(\frac{1}{z+2}\)

Решение №16969: \(\frac{2\sqrt{1+\frac{1}{4}\left ( \sqrt{\frac{1}{t}} -\sqrt{t}\right )^{2}}}{\sqrt{1+\frac{1}{4}\left ( \sqrt{\frac{1}{t}} -\sqrt{t}\right )^{2}-\frac{1}{2}\left ( \sqrt{\frac{1}{t}} -\sqrt{t}\right )}}=\frac{2\sqrt{1+\frac{1}{4}\left ( \frac{1}{t}-2+t \right )}}{\sqrt{1+\frac{1}{4}\left ( \frac{1}{t}-2+t \right )-\frac{1}{2}\left ( \frac{1}{\sqrt{t}}-\sqrt{t} \right )}}=\frac{2\sqrt{1+\frac{1}{4}\cdot \frac{1-2t+t^{2}}{t}}}{\sqrt{1+\frac{1}{4}\cdot \frac{1-2t+t^{2}}{t}-\frac{1}{2}\cdot \frac{1-t}{\sqrt{t}}}}=\frac{\sqrt{\frac{1+2t+t^{2}}{t}}}{\frac{1}{2}\sqrt{\frac{1+2t+t^{2}}{t}}-\frac{1-t}{2\sqrt{t}}}=\frac{\frac{1+t}{\sqrt{t}}}{\frac{1+t}{2\sqrt{t}}-\frac{1-t}{2\sqrt{t}}}=\frac{\frac{1+t}{\sqrt{t}}}{\frac{1+t-1+t}{2\sqrt{t}}}=\frac{1+t}{t}\)

Ответ: \(\frac{1+t}{t}\)

Решение №16970: \(x^{2}+2\left ( k-9 \right )x+\left ( k^{2}+3k+4 \right )=\left ( 2\left ( k-9 \right ) \right )^{2}-4\left ( k^{2}+3k+4 \right )=0;4\left ( k^{2}-18k+81-k^{2}-3k-4 \right )=0;-21k+77=0;k=\frac{77}{21}=\frac{11}{3}\)

Ответ: \(\frac{11}{3}\)

Решение №16971: \(\left ( \frac{\left ( x+\sqrt[3]{2ax^{2}} \right )\left ( 2a+\sqrt[3]{4a^{2}x} \right )^{-1}-1}{\sqrt[3]{x}-\sqrt[3]{2a}}\left ( 2a \right )^{-\frac{1}{3}} \right )^{-6}=\left ( \frac{\frac{x+2^{\frac{1}{3}}a^{\frac{1}{3}}x^{\frac{2}{3}}}{2a+2^{\frac{2}{3}}a^{\frac{2}{3}}x^{\frac{1}{3}}}-1}{x^{\frac{1}{3}}-2^{\frac{1}{3}}a^{\frac{1}{3}}}-\frac{1}{2^{\frac{1}{3}}a^{\frac{1}{3}}} \right )^{-6}=\left ( \frac{x^{\frac{2}{3}}-2^{\frac{2}{3}}a^{\frac{2}{3}}}{2^{\frac{2}{3}}a^{\frac{2}{3}}\left ( x^{\frac{1}{3}-2^{\frac{1}{3}}a^{\frac{1}{3}}} \right )}-\frac{1}{2^{\frac{1}{3}}a^{\frac{1}{3}}} \right )^{-6}=\left ( \frac{x^{\frac{1}{3}}+2^{\frac{1}{3}}a^{\frac{1}{3}}-2^{\frac{1}{3}}a^{\frac{1}{3}}}{2^{\frac{2}{3}}a^{\frac{2}{3}}} \right )^{-6}=\left ( \frac{x^{\frac{1}{3}}}{2^{\frac{2}{3}}a^{\frac{2}{3}}} \right )^{-6}=\frac{16a^{4}}{x^{2}}\)

Ответ: \(\frac{16a^{4}}{x^{2}}\)

Решение №16972: \(\left ( \frac{1+\sqrt{x}}{\sqrt{1+x}}-\frac{\sqrt{1+x}}{1+\sqrt{x}} \right )^{2}-\left ( \frac{1-\sqrt{x}}{\sqrt{1+x}}-\frac{\sqrt{1+x}}{1-\sqrt{x}} \right )^{2}=\left ( \frac{1+2\sqrt{x}+x-1-x}{\sqrt{1+x}\left ( 1+\sqrt{x} \right )} \right )^{2}-\left ( \frac{1-2\sqrt{x}+x-1-x}{\sqrt{1+x}\left ( 1-\sqrt{x} \right )} \right )^{2}=\left ( \frac{2\sqrt{x}}{\sqrt{1+x}\left ( 1+\sqrt{x} \right )} \right )^{2}-\left ( \frac{-2\sqrt{x}}{\sqrt{1+x}\left ( 1-\sqrt{x} \right )} \right )^{2}=\frac{4x}{\left (\sqrt{1+x}\left ( 1+\sqrt{x} \right )\right )^{2}}-\frac{4x}{\left (\sqrt{1+x}\left ( 1-\sqrt{x} \right )\right )^{2}}=\frac{-16x}{\left ( 1+x \right )\left ( 1-x \right )\left ( 1-x \right )}=\frac{16x\sqrt{x}}{\left ( 1-x^{2} \right )\left ( x-1 \right )}\)

Ответ: \(\frac{16x\sqrt{x}}{\left ( 1-x^{2} \right )\left ( x-1 \right )}\)

Решение №16973: \(\left ( \sqrt[3]{\frac{x+1}{x-1}}+\sqrt[3]{\frac{x-1}{x+1}}-2 \right )^{\frac{1}{2}}=\sqrt{\sqrt[3]{\frac{x+1}{x-1}}+\frac{1}{\sqrt[3]{\frac{x+1}{x-1}}}-2}=\left | \sqrt[3]{\frac{x+1}{x-1}}-1 \right |\sqrt[6]{\frac{x-1}{x+1}}=\left | \sqrt[3]{\frac{2a^{3}}{2}}-1 \right |\sqrt[6]{\frac{2}{2a^{3}}}=\left | a-1 \right |\frac{1}{\sqrt{a}}=\frac{1-a}{\sqrt{a}};\frac{a-1}{\sqrt{a}}\)

Ответ: \(\frac{1-a}{\sqrt{a}};\frac{a-1}{\sqrt{a}}\)

Решение №16974: \(\left ( \frac{\sqrt{a}}{2}-\frac{1}{2\sqrt{a}} \right )^{2}\cdot \left ( \frac{\sqrt{a}-1}{\sqrt{a}+1}-\frac{\sqrt{a}+1}{\sqrt{a}-1}\right )=\left ( \frac{\left ( \sqrt{a} \right )^{2}-1}{2\sqrt{a}} \right )\cdot \frac{\left ( \sqrt{a}-1 \right )^{2}-\left ( \sqrt{a}+1 \right )^{2}}{\left ( \sqrt{a}+1 \right )\left ( \sqrt{a}-1 \right )}=\frac{\left ( a-1 \right )^{2}}{4a}\cdot \frac{a-2\sqrt{a}+1-a-2\sqrt{a}-1}{a-1}=\frac{\left ( a-1 \right )^{2}\left ( -4\sqrt{a} \right )}{4a\left ( a-1 \right )}=-\frac{a-1}{\sqrt{a}}=\frac{1-a}{\sqrt{a}}\)

Ответ: \(\frac{1-a}{\sqrt{a}}\)

Решение №16975: \(\frac{\left ( x+1 \right )^{-\frac{1}{2}}}{\left ( x-1 \right )^{-\frac{1}{2}}-\left ( x+1 \right )^{-\frac{1}{2}}}=\frac{1}{\sqrt{x+1}}\cdot \frac{\sqrt{x-1}\sqrt{x+1}}{\sqrt{x+1}-\sqrt{x-1}}=\frac{\sqrt{x-1}\left ( \sqrt{x+1}+\sqrt{x-1} \right )}{\left ( \sqrt{x+1} \right )^{2}-\left ( \sqrt{x-1} \right )^{2}}=\frac{\sqrt{x-1}\left ( \sqrt{x+1}+\sqrt{x-1} \right )}{x+1-x+1}=\frac{\sqrt{x-1}\left ( \sqrt{x+1}+\sqrt{x-1} \right )}{2}=\frac{\sqrt{\frac{a^{2}+1}{2a}-1}\left ( \sqrt{\frac{a^{2}+1}{2a}+1} +\sqrt{\frac{a^{2}+1}{2a}-1}\right )}{2}=\frac{\left | a-1 \right |\left ( a+1 \right )+\left ( a-1 \right )^{2}}{4a}=\frac{1-a}{2a};\frac{a-1}{2}\)

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

Решение №16976: \(\frac{b^{2}-3b-\left ( b-1 \right )\sqrt{b^{2}-4}+1}{b^{2}+3b-\left ( b-1 \right )\sqrt{b^{2}-4}+1}\sqrt{\frac{b+2}{b-2}}=\frac{\left ( b^{2}-3b+b \right )-\left ( b-1 \right )\sqrt{\left ( b-2 \right )\left ( b+2 \right )}}{\left ( b^{2}-3b+b \right )-\left ( b-1 \right )\sqrt{\left ( b-2 \right )\left ( b+2 \right )}{\sqrt{\frac{b+2}{b-2}}=-\frac{b-1}{b+1}=\frac{1-b}{1+b}\)

Ответ: \(\frac{1-b}{1+b}\)

Решение №16977: \(\left ( \left ( \sqrt[4]{p}-\sqrt[4]{q} \right )^{-2} +\left ( \sqrt[4]{p} +\sqrt[4]{q}\right )^{-2}\right ):\frac{\sqrt{p}+\sqrt{q}}{p-q}=\left ( \frac{1}{\left ( \sqrt[4]{p}-\sqrt[4]{q} \right )^{2}} +\frac{1}{\left ( \sqrt[4]{p}+\sqrt[4]{q} \right )^{2}}\right )\cdot \frac{p-q}{\sqrt{p}+\sqrt{q}}=\frac{\left ( \sqrt[4]{p}+\sqrt[4]{q} \right )^{2}+\left ( \sqrt[4]{p}-\sqrt[4]{q} \right )^{2}}{\sqrt{p}-\sqrt{q}}\cdot \frac{\left ( \sqrt{p}-\sqrt{q} \right )\left ( \sqrt{p}+\sqrt{q} \right )}{\sqrt{p}+\sqrt{q}}=\frac{\sqrt{p}+2\sqrt[4]{pq}+\sqrt{q}+\sqrt{p}-2\sqrt[4]{pq}+\sqrt{q}}{\sqrt{p}-\sqrt{q}}=\frac{2\left ( \sqrt{p}+\sqrt{q} \right )}{\sqrt{p}-\sqrt{q}}=\frac{2\left ( \sqrt{p}+\sqrt{q} \right )\left ( \sqrt{p}+\sqrt{q} \right )}{\left ( \sqrt{p}-\sqrt{q} \right )\left ( \sqrt{p}-\sqrt{q} \right )}=\frac{2\left ( \sqrt{p}+\sqrt{q} \right )^{2}}{p-q}\)

Ответ: \(\frac{2\left ( \sqrt{p}+\sqrt{q} \right )^{2}}{p-q}\)

Решение №16978: \(\frac{a}{b^{3}-1}-\frac{b}{a^{3}-1}=\frac{\left ( a^{4}-b^{4} \right )-\left ( a-b \right )}{a^{3}b^{3}-\left ( a^{3}+b^{3} \right )+1}=\frac{\left ( a-b \right )\left ( 1-2ab-1 \right )}{a^{3}b^{3}-\left ( a^{3}+b^{3} \right )+1}=\frac{\left ( a-b \right )\left ( -2ab \right )}{a^{2}b^{3}+3ab}=\frac{2\left ( b-a \right )}{a^{2}b^{2}+3}\)

Ответ: \(\frac{2\left ( b-a \right )}{a^{2}b^{2}+3}\)

Решение №16979: \(\frac{\left ( \sqrt[3]{\left (r^{2}+4}\right )\cdot \sqrt{1+\frac{4}{r^{2}}}-\sqrt[3]{\left ( r^{2}+4 \right )\sqrt{1-\frac{4}{r^{2}}}} \right )^{2}}{r^{2}-\sqrt{r^{4}-16}}=\frac{2\left ( r^{2}-\sqrt[3]{\left ( r^{4}-16 \right )^{\frac{3}{2}}} \right )}{\sqrt[3]{r^{2}}\left ( r^{2}-\sqrt{r^{4}-16} \right )}=\frac{2}{\sqrt[3]{r^{2}}}=\frac{2\sqrt[3]{r}}{r}\)

Ответ: \(\frac{2\sqrt[3]{r}}{r}\)

Решение №16980: \(\sqrt{\frac{2a}{\left ( 1+a \right )\sqrt[3]{1+a}}}\cdot \sqrt[3]{\frac{4+\frac{8}{a}+\frac{4}{a^{2}}}{\sqrt{2}}}=\sqrt[6]{\left ( \frac{2a}{\left ( 1+a \right )\sqrt[3]{1+a}} \right )^{3}}\cdot \sqrt[6]{\left ( \frac{\frac{4+\frac{8}{a}+\frac{4}{a^{2}}}{\sqrt{2}}} \right )}=\sqrt[6]{\frac{8a^{3}}{\left ( 1+a \right )^{4}}\cdot \sqrt[6]{\frac{8\left ( 1+a \right )^{4}}{a^{4}}}}=\sqrt[6]{\frac{64}{a}}=\frac{2\sqrt[6]{a^{5}}}{a}\)

Ответ: \(\frac{2\sqrt[6]{a^{5}}}{a}\)

Решение №16981: \(\frac{z^{3}}{3}-z=\frac{\sqrt[3]{\sqrt{3}+\sqrt{2}}+\sqrt[3]{\sqrt{3}-\sqrt{2}}}{3}-\left ( \sqrt[3]{\sqrt{3}+\sqrt{2}}+\sqrt[3]{\sqrt{3}-\sqrt{2}} \right )=\frac{2\sqrt{3}+3\sqrt[3]{\left ( 3-2 \right )\left ( \sqrt{3}+\sqrt{2} \right )}-3\sqrt[3]{\left ( 3-2 \right )\left ( \sqrt{3}-\sqrt{2} \right )}-3\sqrt[3]{\sqrt{3}+\sqrt{2}}-3\sqrt[3]{\sqrt{3}-\sqrt{2}}}{3}=\frac{2\sqrt{3}+3\sqrt[3]{\sqrt{3}+\sqrt{2}}+3\sqrt[3]{\sqrt{3}-\sqrt{2}}-3\sqrt[3]{\sqrt{3}+\sqrt{2}}-3\sqrt[3]{\sqrt{3}-\sqrt{2}}}{3}=\frac{2\sqrt{3}}{3}; x^{3}+3x=\left ( \sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2} \right )^{3}+3\left ( \sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2} \right )=\sqrt{5}+2-3\sqrt[3]{\left ( \sqrt{5}+2 \right )^{2}\left ( \sqrt{5}-2 \right )}+3\sqrt[3]{\left ( \sqrt{5}+2 \right )\left ( \sqrt{5}-2 \right )^{2}}-\sqrt{5}+2+3\sqrt[3]{\sqrt{5}+2}-3\sqrt[3]{\sqrt{5}-2}=4-3\sqrt[3]{\sqrt{5}+2}+3\sqrt[3]{\sqrt{5}-2}+3\sqrt[3]{\sqrt{5}+2}-3\sqrt[3]{\sqrt{5}-2}=4\)

Ответ: \(\frac{2\sqrt{3}}{3}; 4\)

Решение №16982: \(\frac{6}{\sqrt{2}+\sqrt{3}+\sqrt{5}}=\frac{6\left ( \sqrt{2}+\sqrt{3}-\sqrt{5} \right )}{\left ( \sqrt{2}+\sqrt{3}+\sqrt{5} \right )\left ( \sqrt{2}+\sqrt{3}-\sqrt{5} \right )}=\frac{6\left ( \sqrt{2}+\sqrt{3}-\sqrt{5} \right )}{2+3-5+2\sqrt{2*3}}=\frac{\sqrt{4*3}+\sqrt{9*2}-\sqrt{30}}{2}=\frac{2\sqrt{3}+3\sqrt{2}-\sqrt{30}}{2}\)

Ответ: \(\frac{2\sqrt{3}+3\sqrt{2}-\sqrt{30}}{2}\)

Решение №16983: \(\left ( \left ( 1-p^{2} \right )^{-\frac{1}{2}} -\left ( 1+p^{2} \right )^{-\frac{1}{2}} \right )^{2}+2\left ( 1-p^{4} \right )^{-\frac{1}{2}}=\left ( \frac{1}{\sqrt{1-p^{2}}}-\frac{1}{\sqrt{1+p^{2}}} \right )^{2}+\frac{2}{\sqrt{1-p^{4}}}=\left ( \frac{\sqrt{1+p^{2}}-\sqrt{1-p^{2}}}{\sqrt{1-p^{4}}} \right )^{2}+\frac{2}{\sqrt{1-p^{4}}}=\frac{1+p^{2}-2\sqrt{1-p^{4}}+1-p^{2}}{1-p^{4}}+\frac{2}{\sqrt{1-p^{4}}}=\frac{2-2\sqrt{1-p^{4}}}{1-p^{4}}+\frac{2}{\sqrt{1-p^{4}}}=\frac{2-2\sqrt{1-p^{4}}+2\sqrt{1-p^{4}}}{1-p^{4}}=\frac{2}{1-p^{4}}\)

Ответ: \(\frac{2}{1-p^{4}}\)

Решение №16984: \(\frac{x\left ( x^{2}-a^{2} \right )^{-\frac{1}{2}}+1}{a\left ( x-a \right )^{-\frac{1}{2}}+\left ( x+a \right )^{\frac{1}{2}}}:\frac{a^{2}\sqrt{x+a}}{x-\left ( x^{2}-a^{2} \right )^{\frac{1}{2}}}+\frac{1}{x^{2}-ax}=\frac{\frac{x}{\sqrt{x^{2}-a^{2}}+1}}{\frac{a}{\sqrt{x-a}}+\sqrt{x-a}}:\frac{a^{2}\sqrt{x+a}}{x-\sqrt{x^{2}-a^{2}}}+\frac{1}{x\left ( x-a \right )}=\frac{\left ( x+\sqrt{x^{2}-a^{2}}\sqrt{x-a} \right )}{\sqrt{\left ( x+a \right )\left ( x-a \right )}\left ( a+x-a \right )}\cdot \frac{x-\sqrt{x^{2}-a^{2}}}{a^{2}\sqrt{x+a}}+\frac{1}{x\left ( x-a \right )}=\frac{x^{2}-x^{2}+a^{2}}{a^{2}x\left ( x+a \right )}+\frac{1}{\left ( x-a \right )}=\frac{1}{\left ( x+a \right )}+\frac{1}{\left ( x-a \right )}=\frac{2}{x^{2}-a^{2}}\)

Ответ: \(\frac{2}{x^{2}-a^{2}}\)

Решение №16985: \(\frac{4a^{2}-b^{2}}{a^{6}-8b^{6}}\sqrt{a^{2}-2b\sqrt{a^{2}-b^{2}}}\cdot \frac{a^{4}+2a^{2}b^{2}+4b^{4}}{4a^{2}+4ab+b^{2}}\cdot \sqrt{a^{2}+2b\sqrt{a^{2}-b^{2}}}=\frac{\left ( 2a-b \right )\left ( 2a+b \right )}{\left ( a^{2} \right )^{3}-\left ( 2b^{2} \right )^{3}}\cdot \frac{a^{4}+2a^{2}b^{2}+4b^{4}}{\left ( 2a+b \right )^{2}}\cdot \sqrt{\left (a^{2}-2b\sqrt{a^{2}-b^{2}} \right )\left (a^{2}+2b\sqrt{a^{2}-b^{2}}\right )}=\frac{\left ( 2a-b \right )\left ( a^{4}+2a^{2}b^{2}+4b^{4} \right )}{\left ( a^{2}-2b^{2} \right )\left ( a^{4}+2a^{2}b^{2}+4b^{4} \right )\left ( 2a+b \right )}\cdot \sqrt{a^{4}-4b^{2}\left ( a^{2}-b^{2} \right )}=\frac{\left ( 2a-b \right )\left ( a^{2}-2b^{2} \right )}{\left ( a^{2}-2b^{2} \right )\left ( 2a+b \right )}=\frac{2a-b}{2a+b}=\frac{2\cdot \frac{4}{3}-0.25}{2\cdot \frac{4}{3}+0.25}=\frac{7.25}{8.75}=\frac{29}{35}\)

Ответ: \(\frac{29}{35}\)

Решение №16986: \(\frac{\left ( z-1 \right )\left ( z+2 \right )\left ( z-3 \right )\left ( z+4 \right )}{23}; x=\frac{\sqrt{3}-1}{2};=\frac{\left (\frac{\sqrt{3}-1}{2}-1 \right )\left ( \frac{\sqrt{3}-1}{2}+2 \right )\left ( \frac{\sqrt{3}-1}{2}-3 \right )\left ( \frac{\sqrt{3}-1}{2}+4 \right )}{23}=\frac{\left ( \left ( \frac{\sqrt{3}-1}{2} \right )^{2}+\frac{\sqrt{3}-1}{2}-2 \right )\left ( \left ( \frac{\sqrt{3}-1}{2} \right )^{2}+\frac{\sqrt{3}-1}{2}-12 \right )}{23}=\frac{\left ( \frac{1}{2} \right )^{2}-14\frac{1}{2}+24}{23}=\frac{\frac{1}{4}-7+24}{23}=\frac{3}{4}\)

Ответ: \(\frac{3}{4}\)

Решение №16987: \(\left ( x+1 \right )\left ( x+2 \right )\left ( x+3 \right )\left ( x+4 \right ); x=\frac{\sqrt{7}-5}{2};=\left (\frac{\sqrt{7}-5}{2}+1 \right )\left ( \frac{\sqrt{7}-5}{2}+2 \right )\left ( \frac{\sqrt{7}-5}{2}+3 \right )\left ( \frac{\sqrt{7}-5}{2}+4 \right )=\left ( \left ( \frac{\sqrt{7}-5}{2} \right )^{2}+5\frac{\sqrt{7}-5}{2}+4 \right )\left ( \left ( \frac{\sqrt{7}-5}{2} \right )^{2}+5\frac{\sqrt{7}-5}{2}+6 \right )=\left ( \frac{-9}{2} \right )^{2}+10\left ( \frac{-9}{2}+24 \right )=-\frac{3}{4}\)

Ответ: \(-\frac{3}{4}\)

Решение №16988: \(\frac{\sqrt[4]{7\sqrt[3]{54}+15\sqrt[3]{128}}}{\sqrt[3]{4\sqrt[4]{32}}+\sqrt[3]{9\sqrt[4]{162}}}=\frac{\sqrt[4]{7\cdot 3\sqrt[3]{2}+15\cdot 4\sqrt[3]{2}}}{\sqrt[3]{4\cdot 2\sqrt[4]{2}}+\sqrt[3]{9\cdot 3\sqrt[4]{2}}}=\frac{\sqrt[4]{81\sqrt[3]{2}}}{2\sqrt[3]{\sqrt[4]{2}}+3\sqrt[3]{\sqrt[4]{2}}}=\frac{3\sqrt[12]{2}}{5\sqrt[12]{2}}=\frac{3}{5}\)

Ответ: \(\frac{3}{5}\)

Решение №16989: \(\frac{x\left | x-3 \right |+x^{2}-9}{2x^{3}-3x^{2}-9x}=\frac{x\left | x-3 \right |+x^{2}-9}{x\left ( 2x^{2}-3x-9 \right )}=\frac{-x\left ( x-3 \right )+x^{2}-9}{x\left ( 2x^{2}-3x-9 \right )};\frac{x\left ( x-3 \right )+x^{2}-9}{x\left ( 2x^{2}-3x-9 \right )}= \frac{3\left ( x-3 \right )}{x\left ( 2x+3 \right )\left ( x-3 \right )};\frac{2x^{2}-3x-9}{x\left ( 2x^{2}-3x-9 \right )}=\frac{3}{x\left ( 2x+3 \right )};\frac{1}{x}\)

Ответ: \(\frac{3}{x\left ( 2x+3 \right )};\frac{1}{x}\)

Решение №16990: \(\frac{5\sqrt[3]{4\sqrt[3]{192}}+7\sqrt[3]{18\sqrt[3]{81}}}{\sqrt[3]{12\sqrt[3]{24}}+6\sqrt[3]{375}}=\frac{5\sqrt[3]{4\cdot 4\sqrt[3]{3}}+7\sqrt[3]{18\cdot 3\sqrt[3]{3}}}{\sqrt[3]{12\cdot 2\sqrt[3]{3}}+6\cdot 5\sqrt[3]{3}}=\frac{31\sqrt[3]{2\sqrt[3]{3}}}{3\sqrt[3]{2\sqrt[3]{3}}}=\frac{31}{3}\)

Ответ: \(\frac{31}{3}\)

Решение №16991: \(\frac{1}{\sqrt{x^{2}+4x+4}}+\left | x-2 \right |=\frac{1}{\sqrt{\left ( x+2 \right )^{2}}}+\left | x-2 \right |=\frac{1}{\left | x+2 \right |}+\left | x-2 \right |=\frac{1+\left | x^{2}-4 \right |}{\left | x+2 \right |}=\frac{1+x^{2}-4}{-\left ( x+2 \right )};\frac{1-\left (x^{2}-4 \right )}{ x+2 };\frac{1+x^{2}-4}{ x+2}=\frac{3-x^{2}}{x+2};\frac{5-x^{2}}{x+2};\frac{x^{2}-3}{x+2}\)

Ответ: \(\frac{3-x^{2}}{x+2};\frac{5-x^{2}}{x+2};\frac{x^{2}-3}{x+2}\)

Решение №16992: \(\frac{\sqrt{x-4\sqrt{x-4}}+2}{\sqrt{x-4\sqrt{x-4}}-2}=\frac{\sqrt{x-4-4\sqrt{x-4}+4}+2}{\sqrt{x-4+4\sqrt{x-4}+4}-2}=\frac{\left | \sqrt{x-4}-2 \right |+2}{\sqrt{x-4}+2-2}=\frac{\left | \sqrt{x-4}-2 \right |+2}{\sqrt{x-4}}=\frac{4}{\sqrt{x-4}}-1;1\)

Ответ: \(\frac{4}{\sqrt{x-4}}-1;1\)

Решение №16993: \(\frac{\sqrt{\left ( 1+\frac{b}{a} \right )^{2}-\frac{4b+1}{a}}\left ( \sqrt{a}+\sqrt{b+\sqrt{a}} \right )^{-\frac{1}{2}}}{\sqrt{a-b+\sqrt{a}}\sqrt{\sqrt{a}-\sqrt{b+\sqrt{a}}}}=\frac{\frac{\sqrt{\left ( a-b \right )^{2}-a}}{a}}{\sqrt{\left ( a-b+\sqrt{a} \right )\left ( a-b-\sqrt{a} \right )}}=\frac{\sqrt{\left ( a-b \right )^{2}-a}}{a\sqrt{\left ( a-b \right )^{2}-a}}=\frac{1}{a}=\frac{1}{2.25}=\frac{1}{\frac{9}{4}}=\frac{4}{9}\)

Ответ: \(\frac{4}{9}\)

Решение №16994: \(\frac{4-\sqrt[3]{a^{2}}}{\left ( 2+\sqrt[3]{ab} \right )^{2}-\left ( \sqrt[3]{a}+2\sqrt[3]{b} \right )^{2}}=\frac{4-\sqrt[3]{a^{2}}}{\left ( 4-\sqrt[3]{a^{2}} \right )-\sqrt[3]{b^{2}\left ( 4-\sqrt[3]{a^{2}} \right )}}=\frac{4-\sqrt[3]{a^{2}}}{\left ( 4-\sqrt[3]{a^{2}} \right )\left ( 1-\sqrt[3]{a^{2}} \right )}=\frac{1}{1-\sqrt[3]{a^{2}}}=\frac{1}{1-\sqrt[3]{0.008}}=\frac{1}{1-0.2}=\frac{1}{0.8}=\frac{5}{4}\)

Ответ: \(\frac{5}{4}\)

Решение №16995: \(\sqrt{\frac{x}{x-a^{2}}}:\left ( \frac{\sqrt{x}-\sqrt{x-a^{2}}}{\sqrt{x}+\sqrt{x-a^{2}}}- \frac{\sqrt{x}+\sqrt{x-a^{2}}}{\sqrt{x}-\sqrt{x-a^{2}}}\right )=\sqrt{\frac{x}{x-a^{2}}}:\left ( \frac{\left ( \sqrt{x}-\sqrt{x-a^{2}} \right )^{2}-\left ( \sqrt{x}+\sqrt{x-a^{2}} \right )^{2}}{\left ( \sqrt{x}+\sqrt{x-a^{2}} \right )\left ( \sqrt{x}-\sqrt{x-a^{2}} \right )} \right )=\sqrt{\frac{x}{x-a^{2}}}:\frac{-4\sqrt{x\left ( x-a^{a} \right )}}{x-x+a^{2}}=-\frac{a^{2}}{4\left ( x-a^{2} \right )}=\frac{a^{2}}{4\left ( a^{2}-x \right )}\)

Ответ: \(\frac{a^{2}}{4\left ( a^{2}-x \right )}\)

Решение №16996: \(\frac{2a^{4}+a^{3}+4a^{2}+a+2}{2a^{3}-a^{2}+a-2}=\frac{\left ( 2a^{4}+2a^{2} \right )+\left ( a^{3}+a \right )+\left ( 2a^{2}+2 \right )}{\left ( 2a^{3}-2a^{2} \right )+\left ( a^{2}-a \right )+\left ( 2a-2 \right )}=\frac{\left ( a^{2}+1 \right )\left ( 2a^{2}+a+2 \right )}{\left ( a-1 \right )\left ( 2a^{2}+a+2 \right )}=\frac{a^{2}+1}{a-1}\)

Ответ: \(\frac{a^{2}+1}{a-1}\)

Решение №16997: \(\frac{a^{3}+a^{2}-2a}{a\left | a+2 \right |-a^{2}+4}=\frac{a\left ( a^{2}-1 \right )+a\left ( a-1 \right )}{a\left | a+2 \right |-a^{2}+4}=\frac{a\left ( a+2 \right )\left ( a-1 \right )}{a\left | a+2 \right |-\left ( a-2 \right )\left ( a+2 \right )}=\frac{+}{}\frac{a\left ( a+2 \right )\left ( a-1 \right )}{a\left ( a+2 \right )-\left ( a-2 \right )\left ( a+2 \right )}=\frac{+}{}\frac{a\left ( a+2 \right )\left ( a-1 \right )}{\left ( a+2 \right )\left ( a-a+2 \right )}=-\frac{a}{2};\frac{a\left ( a-1 \right )}{2}\)

Ответ: \(-\frac{a}{2};\frac{a\left ( a-1 \right )}{2}\)

Решение №16998: \(\left ( \sqrt{ab}-ab\left ( a+\sqrt{ab} \right )^{-1} \right ):\left ( 2\left (\left ( ab \right )^{\frac{1}{2}}-b \right )\left ( a-b \right )^{-1}\right )=\left ( \sqrt{ab}-\frac{ab}{a+\sqrt{ab}} \right ):\frac{2\left ( \sqrt{ab}-b \right )}{a-b}=\left ( \sqrt{ab}-\frac{ab}{\sqrt{a}\left ( \sqrt{a}+\sqrt{b} \right )} \right ):\frac{2\sqrt{b}\left ( \sqrt{a}-\sqrt{b} \right )}{\left ( \sqrt{a}-\sqrt{b} \right )\left ( \sqrt{a}+\sqrt{b} \right )}=\sqrt{ab}\left ( 1-\frac{\sqrt{b}}{\sqrt{a}+\sqrt{b}} \right )\cdot \frac{\sqrt{a}+\sqrt{b}}{2\sqrt{b}}=\frac{a}{2}\)

Ответ: \(\frac{a}{2}\)

Решение №16999: \(\frac{a+1}{2\sqrt[3]{\sqrt{3}-\sqrt{2}}\sqrt[6]{5+2\sqrt{6}}+\frac{1}{a}+a}=\frac{\left ( a+1 \right )a}{2a\sqrt[3]{\sqrt{3}-\sqrt{2}}\sqrt[3]{\sqrt{3}+\sqrt{2}}+1+a^{2}}=\frac{\left ( a+1 \right )a}{2a\sqrt[3]{3-2}+1+a^{2}}=\frac{\left ( a+1 \right )a}{\left ( a+1 \right )^{2}}=\frac{a}{a+1}\)

Ответ: \(\frac{a}{a+1}\)

Решение №17000: \(\frac{\sqrt{2a+2\sqrt{a^{2}-9}}}{\sqrt{2a-2\sqrt{a^{2}-9}}}=\frac{\sqrt{a+3+2\sqrt{\left ( a+3 \right )\left ( a-3 \right )}+a-3}}{\sqrt{a+3-2\sqrt{\left ( a+3 \right )\left ( a-3 \right )}+a-3}}=\frac{\sqrt{a+3}+\sqrt{a-3}}{\sqrt{a+3}-\sqrt{a-3}}=\frac{\left ( \sqrt{a+3}+\sqrt{a-3} \right )^{2}}{\left ( \sqrt{a+3} \right )^{2}-\left ( \sqrt{a-3} \right )^{2}}=\frac{2a+2\sqrt{a^{2}-9}}{6}=\frac{a+\sqrt{a^{2}-9}}{3}\)

Ответ: \(\frac{a+\sqrt{a^{2}-9}}{3}\)

Решение №17001: \(\frac{a^{4}-a^{2}-2a-1}{a^{3}-2a^{2}+1}:\frac{a^{4}+2a^{3}-a-2}{1+\frac{4}{a}+\frac{4}{a^{2}}}=\frac{a^{2}+a+1}{a-1}:\frac{\left ( a+2 \right )\left ( a^{3}-1 \right )a^{2}}{\left ( a+2 \right )^{2}}=\frac{a^{2}+a+1}{a-1}:\frac{\left ( a-1 \right )\left ( a^{2}+a+1 \right )a^{2}}{a+2}=\frac{a^{2}+a+1}{a-1}\cdot \frac{a+2}{\left ( a-1 \right )\left ( a^{2}+a+1 \right )a^{2}}=\frac{a+2}{a^{2}\left ( a-1 \right )^{2}}\)

Ответ: \(\frac{a+2}{a^{2}\left ( a-1 \right )^{2}}\)