Решение №17007: \(\frac{\sqrt{z^{2}-1}}{\sqrt{z^{2}-1}-z}=\frac{z^{2}-1+z\sqrt{z^{2}-1}}{z^{2}-1-z^{2}}=-\left ( z^{2}+z\sqrt{z^{2}-1}-1 \right )=1-z^{2}-z\sqrt{z^{2}-1}=1-\left ( \frac{1}{2}\left ( \sqrt{m}+\frac{1}{\sqrt{m}} \right ) \right )^{2}-\frac{1}{2}\left ( \sqrt{m}+\frac{1}{\sqrt{m}} \right )\sqrt{\left ( \frac{1}{2}\left ( \sqrt{m}+\frac{1}{\sqrt{m}} \right ) \right )^{2}-1}=\frac{4m-m^{2}-2m-1}{4m}-\frac{m+1}{2\sqrt{m}}\sqrt{\frac{m^{2}-2m+1}{4m}}=\frac{-\left ( m-1 \right )^{2}-\left ( m+1 \right )\left | m-1 \right |}{4m}=\frac{m-1}{2m};\frac{1-m}{2}\)

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

Решение №17008: \(\frac{\sqrt[3]{m+4\sqrt{m-4}}\sqrt[3]{\sqrt{m-4}+2}}{\sqrt[3]{m-4\sqrt{m-4}}\sqrt[3]{\sqrt{m-4}-2}}\cdot \frac{m-4\sqrt{m-4}}{2}=\frac{\sqrt[3]{\left ( \sqrt{m-4}+2 \right )^{3}}}{\sqrt[3]{\left ( \sqrt{m-4}-2 \right )^{3}}}\cdot \frac{\left ( \sqrt{m-4}-2 \right )^{2}}{2}=\frac{\left ( \sqrt{m-4}+2 \right )\left ( \sqrt{m-4}-2 \right )^{2}}{2\left ( \sqrt{m-4}-2 \right )}=\frac{\left ( \left ( \sqrt{m-4} \right )^{2}-2^{2} \right )}{2}=\frac{m-4-4}{2}=\frac{m-8}{2}\)

Ответ: \(\frac{m-8}{2}\)

Решение №17009: \(\frac{1}{n^{2}+3n+2}=-\frac{1}{n+2}+\frac{1}{n+1};\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{n^{2}+3n+2}=\frac{1}{2}-\frac{1}{n+2}=\frac{n+2-2}{2\left ( n+2 \right )}=\frac{n}{2n+4}\)

Ответ: \(\frac{n}{2n+4}\)

Решение №17010: \(frac{1}{n\left ( n+1 \right )}=\frac{1}{n}-\frac{1}{n+1};1-\frac{1}{n+1}=\frac{n+1-1}{n+1}=\frac{n}{n+1}\)

Ответ: \(\frac{n}{n+1}\)

Решение №17011: \(\frac{\left | r-1 \right |\cdot \left | r \right |}{r^{2}-r+1-\left | r \right |}=\frac{-\left ( r-1 \right )\cdot \left (- r \right )}{r^{2}-r+1+ r };\frac{-\left ( r-1 \right ) r }{r^{2}-r+1- r };\frac{\left ( r-1 \right )r}{r^{2}-r+1-r }=\frac{r^{2}-r}{r^{2}+r};\frac{r}{1-r};\frac{r}{r-1}\)

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

Решение №17012: \(\frac{x^{\frac{1}{6}}-y^{\frac{1}{6}}}{x^{\frac{1}{2}}+x^{\frac{1}{3}}y^{\frac{1}{6}}}\cdot \frac{\left ( x^{\frac{1}{3}}+y^{\frac{1}{3}}-4\sqrt[3]{xy} \right )}{x^{\frac{5}{6}}y^{\frac{1}{3}}-x^{\frac{1}{2}}y^{\frac{2}{3}}}+2x^{-\frac{2}{3}}y^{-\frac{1}{6}}=\frac{x^{\frac{2}{6}}+y^{\frac{2}{6}}}{x^{\frac{5}{6}}y^{\frac{2}{6}}}=\frac{x^{\frac{1}{3}}+y^{\frac{1}{3}}}{\sqrt[6]{x^{5}y^{2}}}\)

Ответ: \(\frac{x^{\frac{1}{3}}+y^{\frac{1}{3}}}{\sqrt[6]{x^{5}y^{2}}}\)

Решение №17013: \(\frac{\left ( x^{\frac{2}{m}}-9x^{\frac{2}{n}} \right )\left ( \sqrt[m]{x^{1-m}}-3\sqrt[n]{x^{1-n}} \right )}{\left ( x^{\frac{1}{m}}+3x^{\frac{1}{n}} \right )^{2}-12x^{\frac{m+n}{mn}}}=\frac{\left ( x^{\frac{1}{m}}-3x^{\frac{1}{n}} \right )\left ( x^{\frac{1}{m}}+3x^{\frac{1}{n}} \right )\left ( x^{\frac{1}{m}-1}-3x^{\frac{1}{n}-1} \right )}{x^{\frac{2}{m}}-6x^{\frac{1}{m}+\frac{1}{n}}+9x\frac{2}{n}}=\frac{\left ( x^{\frac{1}{m}}-3x^{\frac{1}{n}} \right )\left ( x^{\frac{1}{m}}+3x^{\frac{1}{n}} \right )\frac{1}{x}\left ( x^{\frac{1}{m}}-3x^{\frac{1}{n}} \right )}{\left ( x^{\frac{1}{m}}-3x^{\frac{1}{n}} \right )^{2}}=\frac{x^{\frac{1}{m}}+3x^{\frac{1}{n}}}{x}\)

Ответ: \(\frac{x^{\frac{1}{m}}+3x^{\frac{1}{n}}}{x}\)

Решение №17014: \(\left ( \frac{bx+4+\frac{4}{bx}}{2b+\left ( b^{2}-4 \right )x-2bx^{2}}+\frac{\left ( 4x^{2}-b^{2} \right )\cdot \frac{1}{b}}{\left ( b+2x \right )^{2}}-8bx \right )\frac{bx}{2}=\left ( -\frac{bx+2}{\left ( 2x-b \right )bx}+\frac{2x+b}{\left ( 2x-b \right )b} \right )\frac{bx}{2}=-\frac{bx+2}{2\left ( 2x-b \right )}+\frac{\left ( 2x+b \right )x}{2\left ( 2x-b \right )}=\frac{x^{2}-1}{2x-b}\)

Ответ: \(\frac{x^{2}-1}{2x-b}\)

Решение №17015: \(\frac{\left (x^{2}-3x+2 \right )^{-\frac{1}{2}}-\left ( x^{2}+3x+2 \right )^{-\frac{1}{2}}}{\left (x^{2}-3x+2 \right )^{-\frac{1}{2}}+\left ( x^{2}+3x+2 \right )^{-\frac{1}{2}}}-1+\frac{\left ( x^{4}-5x^{2}+4 \right )^{\frac{1}{2}}}{3x}=\sqrt{x^{4}-5x^{2}+4}\frac{\sqrt{x^{2}-3x+2}-\sqrt{x^{2}+3x+2}+\sqrt{x^{2}+3x+2}}{3x\sqrt{x^{2}+3x+2}}=\frac{\sqrt{\left ( x^{2}-3x+2 \right )^{2}}}{3x}=\frac{x^{2}-3x+2}{3}\)

Ответ: \(\frac{x^{2}-3x+2}{3}\)

Решение №17016: \(\left ( \frac{x^{4}+5x^{3}+15x-9}{x^{6}+3x^{4}}+\frac{9}{x^{4}} \right ):\frac{x^{3}-4x+3x^{2}-12}{x^{5}}=\frac{x^{4}+5x^{3}+15x-9+9\left ( x^{2}+3 \right )}{x^{4}\left ( x^{2}+3 \right )}:\frac{\left ( x^{2}-4 \right )\left ( x+3 \right )}{x^{5}}=\frac{\left ( x+3 \right )\left ( x+2 \right )\left ( x^{2}+3 \right )}{x^{4}\left ( x^{2}+3 \right )}\cdot \frac{x^{5}}{\left ( x-2 \right )\left ( x+2 \right )\left ( x+3 \right )}=\frac{x}{x-2}\)

Ответ: \(\frac{x}{x-2}\)