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: \(\frac{1-b}{1+b}\)