Решение №17002: \(\frac{a^{3}-2a^{2}+5a+26}{a^{3}-5a^{2}+17a-13}=\frac{\left ( a^{3}+2a^{2} \right )-\left ( 4a^{2}+8a \right )+\left ( 13a+26 \right )}{\left ( a^{3}-a^{2} \right )-\left ( 4a^{2}-8a \right )+\left ( 13a-13 \right )}=\frac{\left ( a+2 \right )\left ( a^{2}-4a+13 \right )}{\left ( a-1 \right )\left ( a^{2}-4a+13 \right )}=\frac{a+2}{a-1}\)

Ответ: \(\frac{a+2}{a-1}\)

Решение №17003: \(\left ( \frac{x+2b}{x-2b}+\frac{x+2a}{x-2a} \right ):\frac{x}{2}; x=\frac{4ab}{a+b};=\left (\frac{\frac{4ab}{a+b}+2b}{\frac{4ab}{a+b}-2b}+\frac{\frac{4ab}{a+b}+2a}{\frac{4ab}{a+b}-2a} \right ):\frac{4ab}{2\left ( a+b \right )}=\left ( \frac{4ab+2ab+b^{2}}{a+b}:\frac{4ab-2ab-2b^{2}}{a+b}+\frac{4ab+2a^{2}+2ab^{2}}{a+b} :\frac{4ab-2a^{2}-2ab^{2}}{a+b}\right )\frac{a+b}{2ab}=\left ( \frac{3a+b}{a-b}+\frac{3b+a}{b-a} \right )\frac{a+b}{2ab}=\frac{2a-2b}{a-b}\cdot \frac{a+b}{2ab}=\frac{a+b}{ab}\)

Ответ: \(\frac{a+b}{ab}\)

Решение №17004: \(\frac{\frac{\left | b-1 \right |}{b}+b\left | b-1 \right |+2-\frac{2}{b}}{\sqrt{b-2+\frac{1}{b}}}=\frac{\frac{\left | b-1 \right |+b^{2}\left | b-1 \right |+2b-2}{b}}{\sqrt{\frac{b^{2}-2b+1}{b}}}=\frac{\left | b-1 \right |\left ( b^{2}+1 \right )+2\left ( b-1 \right )}{b\sqrt{\frac{\left ( b-1 \right )^{2}}{b}}}=\frac{\left | b-1 \right |\left ( b^{2}+1 \right )+2\left ( b-1 \right )}{\left | b-1 \right |\sqrt{b}}=\frac{b^{2}-1}{\sqrt{b}};\frac{b^{2}+3}{\sqrt{b}}\)

Ответ: \(\frac{b^{2}-1}{\sqrt{b}};\frac{b^{2}+3}{\sqrt{b}}\)

Решение №17005: \(\frac{\frac{a-b}{2a-b}-\frac{a^{2}+b^{2}+a}{2a^{2}+ab-b^{2}}}{\left ( 4b^{4}+4ab^{2}+a^{2} \right ):\left ( 2b^{2}+a \right )}\cdot \left ( b^{2}+b+ab+a \right )=\frac{\frac{a-b}{2a-b}-\frac{a^{2}+b^{2}+a}{\left ( a+b \right )\left ( 2a-b \right )}}{\frac{\left ( 2b^{2}+a \right )^{2}}{2b^{2}+a}}\cdot \left ( b^{2}+b+ab+a \right )=\frac{a^{2}-b^{2}-a^{2}-b^{2}-a}{\left ( a+b \right )\left ( 2a-b \right )\left ( 2b^{2}+a \right )}\cdot \left ( b\left ( b+1 \right )+a\left ( b+1 \right ) \right )=\frac{-\left ( b+1 \right )}{2a-b}=\frac{b+1}{b-2a}\)

Ответ: \(\frac{b+1}{b-2a}\)

Решение №17006: \(\frac{\sqrt{\frac{m+2}{m-2}}+\sqrt{\frac{m-2}{m+2}}}{\sqrt{\frac{m+2}{m-2}}-\sqrt{\frac{m-2}{m+2}}}=\frac{\left ( \sqrt{\frac{m+2}{m-2}}+\sqrt{\frac{m-2}{m+2}} \right )^{2}}{\left ( \sqrt{\frac{m+2}{m-2}} \right )^{2}-\left ( \sqrt{\frac{m-2}{m+2}} \right )^{2}}=\frac{\left ( m+2+m-2 \right )^{2}}{4m+4m}=\frac{4m^{2}}{8m}=\frac{m}{2}\)

Ответ: \(\frac{m}{2}\)

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

Решение №17017: \(\frac{x^{2}-1+\left | x+1 \right |}{\left | x \right |\left ( x-2 \right )}=\frac{x^{2}-1-\left ( x+1 \right )}{-x \left ( x-2 \right )};\frac{x^{2}-1+\left ( x+1 \right )}{-x \left ( x-2 \right )};\frac{x^{2}-1+\left ( x+1 \right )}{x \left ( x-2 \right )}=\frac{\left ( x-2 \right )\left ( x+1 \right )}{-x\left ( x-2 \right )};\frac{x\left ( x+1 \right )}{-x\left ( x-2 \right )};\frac{x\left ( x+1 \right )}{x\left ( x-2 \right )}=-\frac{x+1}{x};\frac{x+1}{2-x};\frac{x+1}{x-2}\)

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

Решение №17018: \(\frac{2x-x\left | x-1 \right |+x\left | x \right |+3}{\left | x \right |+x^{2}}=\frac{2x+x\left ( x-1 \right )-x^{2}+3}{-x +x^{2}};\frac{2x+x\left ( x-1 \right )+x^{2}+3}{x +x^{2}};\frac{2x-x\left ( x-1 \right )+x^{2}+3}{x +x^{2}}=\frac{x+3}{x^{2}-x};\frac{2x^{2}+x+3}{x^{2}+x};\frac{3}{x}\)

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

Решение №17019: \(\frac{x^{3}+5x^{2}+3x-9}{x^{3}+x\sqrt{2}-5x+3}=\frac{\left ( x^{2}+2x^{2}-3x \right )+\left ( 3x^{2}+6x-9 \right )}{\left ( x^{3}+2x^{2}-3x \right )-\left ( x^{2}+2x-3 \right )}=\frac{\left ( x^{2}+2x-3 \right )\left ( x+3 \right )}\left ({x^{2}+2x-3} \right )\left ( x-1 \right )=\frac{x+3}{x-1}\)

Ответ: \(\frac{x+3}{x-1}\)

Решение №17020: \(\frac{\frac{x+y}{\sqrt{x}-\sqrt{y}}-\frac{x-y}{\sqrt{x}+\sqrt{y}}}{\frac{\sqrt{x}-\sqrt{y}}{x+y}+\frac{\sqrt{x}+\sqrt{y}}{x-y}}\cdot \frac{y-\sqrt{xy}+x}{2\sqrt{xy}}=\frac{\frac{\left ( x+y \right )\left ( \sqrt{x}+\sqrt{y} \right )-\left ( x-y \right )\left ( \sqrt{x}-\sqrt{y} \right )}{\left ( \sqrt{x}-\sqrt{y} \right )\left ( \sqrt{x}+\sqrt{y} \right )}}{\frac{\left ( \sqrt{x}-\sqrt{y} \right )\left ( x-y \right )+\left ( \sqrt{x}+\sqrt{y} \right )\left ( x+y \right )}{\left ( x+y \right )\left ( x-y \right )}}\cdot \frac{y-\sqrt{xy}+x}{2\sqrt{xy}}=\frac{2\sqrt{xy}\left ( \sqrt{x}+\sqrt{y} \right )}{\left ( \sqrt{x}-\sqrt{y} \right )\left ( \sqrt{x}+\sqrt{y} \right )}\cdot \frac{\left ( x+y \right )\left ( x-y \right )}{2\left ( x\sqrt{x}+y\sqrt{y} \right )} \cdot \frac{y-\sqrt{xy}+x}{2\sqrt{xy}}=\frac{2\sqrt{xy}}{\sqrt{x}-\sqrt{y}}\cdot \frac{\left ( x+y \right )\left ( \sqrt{x}-\sqrt{y} \right )\left ( \sqrt{x}+\sqrt{y} \right )}{2\left ( \sqrt{x}+\sqrt{y} \right )\left ( x-\sqrt{xy}+y \right )}\cdot \frac{y-\sqrt{xy}+x}{2\sqrt{xy}}=\frac{x+y}{2}\)

Ответ: \(\frac{x+y}{2}\)

Решение №17021: \(\left ( \frac{\sqrt{\left ( z+2 \right )^{2}-8z}}{z+2}+\frac{\left ( z-1 \right )^{2}+3 }{z^{3}+8}\right ):\frac{z^{2}-3z+2}{z^{3}-2z^{2}-4z+8}=\left ( \frac{\sqrt{\left ( z-2 \right )^{2}}}{z+2}+\frac{1}{z+2} \right )\frac{\left ( z+2 \right )\left ( z-2 \right )}{z-1}=\frac{\left | z-2 \right |+1}{z+2}\cdot \frac{\left ( z+2 \right )\left ( z-2 \right )}{z-1}=\frac{\left ( \left | z-2 \right |+1 \right )\left ( z-2 \right )}{z-1}=\frac{z^{2}-5z+6}{1-z};z-2\)

Ответ: \(\frac{z^{2}-5z+6}{1-z};z-2\)

Решение №17022: \(\frac{\left ( m^{2}-\frac{1}{n^{2}} \right )^{m}\cdot \left ( n+\frac{1}{m} \right )^{n-m}}{\left ( n^{2}-\frac{1}{m^{2}} \right )^{n}\cdot \left ( m-\frac{1}{n} \right )^{m-n}}=\frac{\left ( \frac{m^{2}n^{2}-1}{n^{2}} \right )^{m}\left ( \frac{mn+1}{m} \right )^{n-m}}{\left ( \frac{m^{2}n^{2}-1}{m^{2}} \right )^{n}\left ( \frac{mn-1}{n} \right )^{m-n}}=\frac{\left ( mn-1 \right )^{m}\left ( mn+1 \right )^{n}m^{2n}n^{m-n}}{mn+1 \right )^{n}m^{n-m}n^{2m}}=\frac{m^{2n}n^{m}m^{m}}{m^{n}n^{2m}n^{n}}=\frac{m^{n}m^{m}}{n^{m}n^{n}}=\frac{m^{m+n}}{n^{m+n}}=\left ( \frac{m}{n} \right )^{m+n}\)

Ответ: \(\left ( \frac{m}{n} \right )^{m+n}\)

Решение №17023: \(\frac{4}{\sqrt[4]{13}+\sqrt[4]{9}}=\frac{4\left ( \sqrt[4]{13^{3}}+\sqrt[4]{13^{2}*9}+\sqrt[4]{13*9^{2}}+\sqrt[4]{9^{3}} \right )}{\left ( \sqrt[4]{13}-\sqrt[4]{9} \right )\left ( \sqrt[4]{13^{2}*9}+\sqrt[4]{13*9^{2}}+\sqrt[4]{9^{3}} \right )}=\frac{4\left ( \sqrt[4]{13}+\sqrt[4]{9} \right )\left ( \sqrt[4]{13^{2}}+\sqrt[4]{9^{2}} \right )}{13-9}=\left ( \sqrt[4]{13}+\sqrt[4]{9} \right )\left ( \sqrt{13}+3\right )\)

Ответ: \(\left ( \sqrt[4]{13}+\sqrt[4]{9} \right )\left ( \sqrt{13}+3\right )\)

Решение №17024: \(\sqrt[4]{\frac{x}{32}}\cdot \frac{\left ( \sqrt[8]{x}-\sqrt[8]{2} \right )^{2}+\left ( \sqrt[8]{x}-\sqrt[8]{2} \right )^{2}}{\sqrt{x}-\sqrt[4]{2x}}:\frac{\left ( \sqrt[4]{x}+\sqrt[4]{2}-\sqrt[8]{2x} \right )\left ( \sqrt[4]{x}+\sqrt[4]{2}+\sqrt[8]{2x} \right )}{2-\sqrt[4]{2x^{3}}}=\sqrt[8]{\frac{x^{2}}{2^{10}}}\cdot \frac{2\left ( \sqrt[8]{x^{2}}+\sqrt[8]{2^{2}} \right )}{\sqrt[8]{x^{2}}\left ( \sqrt[8]{x^{2}}-\sqrt[8]{2^{2}} \right )}:\frac{1}{-\sqrt[8]{2^{2}}\left ( \sqrt[8]{x^{2}}*\sqrt[8]{2^{2}} \right )}=\frac{\sqrt[8]{x^{2}}\sqrt[8]{2^{2}}\left ( \sqrt[8]{x^{2}}+\sqrt[8]{2^{2}} \right )\left ( -\sqrt[8]{2^{2}} \right )\left ( \sqrt[8]{x^{2}}-\sqrt[8]{2^{2}} \right )}{\sqrt[8]{2^{10}}\sqrt[8]{x^{2}}\left ( \sqrt[8]{x^{2}}-\sqrt[8]{2^{2}} \right )}=-\left ( \sqrt[8]{x^{2}}+\sqrt[8]{2^{2}} \right )=-\left ( \sqrt[4]{x}+\sqrt[4]{2} \right )\)

Ответ: \(-\left ( \sqrt[4]{x}+\sqrt[4]{2} \right )\)

Решение №17025: \(\frac{\sqrt{x^{3}}+\sqrt{xy^{2}}-\sqrt{x^{2}y}-\sqrt{y^{3}}}{\sqrt[4]{x^{5}}+\sqrt[4]{x^{4}y}-\sqrt[4]{xy^{4}}-\sqrt[4]{x^{5}}}=\frac{\left ( \sqrt{x^{3}}+\sqrt{xy^{2}} \right )-\left ( \sqrt{x^{2}y}+\sqrt{y^{3}} \right )}{\left ( \sqrt[4]{x^{5}}+\sqrt[4]{x^{4}y} \right )-\left ( \sqrt[4]{xy^{4}}+\sqrt[4]{x^{5}} \right )}=\frac{\sqrt{x}\left ( x+y \right )-\sqrt{y}\left ( x+y \right )}{\sqrt[4]{y}\left ( x+y \right )-\sqrt[4]{x}\left ( x+y \right )}=\frac{\left ( x+y \right )\left ( \sqrt{x}-\sqrt{y} \right )}{\left ( x+y \right )\left ( \sqrt[4]{y}-\sqrt[4]{x} \right )}=-\left ( \sqrt[4]{x}+\sqrt[4]{y} \right )\)

Ответ: \(-\left ( \sqrt[4]{x}+\sqrt[4]{y} \right )\)

Решение №17026: \(\frac{\left ( \sqrt[4]{m}+\sqrt[4]{n} \right )^{2}+\left ( \sqrt[4]{m}-\sqrt[4]{n} \right )^{2}}{2\left ( m-n \right )}:\frac{1}{\sqrt{m^{3}}-\sqrt{n^{3}}}-3\sqrt{mn}=\frac{\sqrt{m}+2\sqrt[4]{mn}+\sqrt{n}+\sqrt{m}-2\sqrt[4]{mn}+\sqrt{n}}{2\left ( \sqrt{m}-\sqrt{n} \right )\left ( \sqrt{m}+\sqrt{n} \right )}\cdot \frac{\left ( \sqrt{m}-\sqrt{n} \right )\left ( \left ( \sqrt{m} \right )^{2}+\sqrt{mn}+\left ( \sqrt{n} \right )^{2} \right )}{1}-3\sqrt{mn}=\frac{2\left ( \sqrt{m}+\sqrt{n} \right )\left ( \sqrt{m}-\sqrt{n} \right )\left ( \left ( \sqrt{m} \right )^{2}+\sqrt{mn}+\left ( \sqrt{n} \right )^{2} \right )}{2\left ( \sqrt{m}-\sqrt{n} \right )\left ( \sqrt{m}+\sqrt{n} \right )}-3\sqrt{mn}=\left ( \sqrt{m} \right )^{2}+\sqrt{mn}+\left ( \sqrt{n} \right )^{2} -3\sqrt{mn}=\left ( \sqrt{m} \right )^{2}-2\sqrt{mn}+\left ( \sqrt{n} \right )^{2}=\left ( \sqrt{m}-\sqrt{n} \right )^{2}\)

Ответ: \(\left ( \sqrt{m}-\sqrt{n} \right )^{2}\)

Решение №17027: \(\left ( \frac{a}{b}\sqrt[3]{b-\frac{4a^{6}}{b^{3}}}-a^{2}\sqrt[3]{\frac{b}{a^{6}}-\frac{4}{b^{3}}}+\frac{2}{ab}\sqrt[3]{a^{3}b^{4}-4a^{9}} \right ):\frac{\sqrt[3]{b^{2}-2a^{3}}}{b^{2}}=\left ( \frac{a}{b}\sqrt[3]{\frac{b^{4}-4a^{6}}{^{3}}}-a^{2}\sqrt[3]{\frac{b^{4}-4a^{6}}{a^{6}b^{3}}}+\frac{2}{ab}\sqrt[3]{a^{3}\left ( b^{4}-4a^{6} \right )} \right )\cdot \frac{b^{2}}{\sqrt[3]{b^{2}-2a^{3}}}=\left ( \frac{a\sqrt[3]{b^{4}-4a^{6}}}{b^{2}} -\frac{a^{2}\sqrt[3]b^{4}-4a^{6}}{{a^{2}b}}+\frac{2a\sqrt[3]{b^{4}-4a^{6}}}{ab}\right )\cdot \frac{b^{2}}{\sqrt[3]{b^{2}-2a^{3}}}=\frac{\sqrt[3]{\left ( b^{2}-2a^{3} \right )\left ( b^{2}+2a^{3} \right )\left ( a+b \right )}}{b^{2}}\cdot \frac{b^{2}}{\sqrt[3]{b^{2}-2a^{3}}}=\left ( a+b \right )\sqrt[3]{b^{2}+2a^{3}}\)

Ответ: \(\left ( a+b \right )\sqrt[3]{b^{2}+2a^{3}}\)

Решение №17028: \(4ab+\frac{\left ( 1+\left ( \frac{a}{b} \right )^{-3} \right )a^{3}}{\left ( \sqrt{a}+\sqrt{b} \right )^{2}-2\sqrt{ab}}-\frac{\left ( \frac{\sqrt{a}+\sqrt{b}}{2b\sqrt{a}} \right )^{-1}+\left ( \frac{\sqrt{a}+\sqrt{b}}{2a\sqrt{b}} \right )^{-1}}{\left ( \frac{a+\sqrt{ab}}{2} \right )^{-1}+\left ( \frac{b+\sqrt{ab}}{2} \right )^{-1}}=4ab+\frac{a^{3}+b^{3}}{a+b}-\frac{\frac{2b\sqrt{a}+2a\sqrt{b}}{\sqrt{a}+\sqrt{b}}}{\frac{2b+2\sqrt{ab}+2a+2\sqrt{ab}}{\left ( a+\sqrt{ab} \right )\left ( b+\sqrt{ab} \right )}}=4ab+a^{2}-ab+b^{2}-\sqrt{ab}\frac{\sqrt{ab}\left ( \sqrt{a}+\sqrt{b} \right )}{\left ( \sqrt{a}+\sqrt{b} \right )^{2}}=a^{2}+3ab+b^{2}-ab=a^{2}+2ab+b^{2}=\left ( a+b \right )^{2}\)

Ответ: \(\left ( a+b \right )^{2}\)

Решение №17029: \(\frac{m^{5}+m^{4}\sqrt[3]{2}+\sqrt[3]{4m^{9}}}{\left | m^{3}-1 \right |-1}=\frac{m^{5}+\sqrt[3]{2}m^{4}+\sqrt[3]{2^{2}}m^{3}}{\left | m^{3}-1 \right |-1}=\frac{m^{3}\left ( m^{2}+\sqrt[3]{2}m+\sqrt[3]{2^{2}} \right )}{\left | m^{3}-1 \right |-1}=-\left ( m^{2}+\sqrt[3]{2}m+\sqrt[3]{2^{2}} \right );\frac{m^{3}}{m-\sqrt[3]{2}}\)

Ответ: \(-\left ( m^{2}+\sqrt[3]{2}m+\sqrt[3]{2^{2}} \right );\frac{m^{3}}{m-\sqrt[3]{2}}\)

Решение №17030: \(\frac{x^{4}-x^{3}-x+1}{x^{3}-5x^{2}+7x-3}\left | x-3 \right |=\frac{\left ( x-1 \right )^{2}\left ( x^{2}+x+1 \right )}{\left ( x-1 \right )^{2}\left ( x-3 \right )}\left | x-3 \right |=\frac{\left ( x^{2}+x+1 \right )\left | x-3 \right |}{x-3}=-\left ( x^{2}+x+1 \right ); x^{2}+x+1 \)

Ответ: \(-\left ( x^{2}+x+1 \right ); x^{2}+x+1\)

Решение №17031: \(x^{8}-16=\left ( x^{4}-4 \right )\left ( x^{4}+4 \right )=\left ( x^{2}-2 \right )\left ( x^{2}+2 \right )\left ( x^{4}+4x^{2}+4-4x^{2} \right )=\left ( x^{2}-2 \right )\left ( x^{2}+2 \right )\left ( \left ( x^{2}+2 \right )^{2} -4x^{2}\right )=\left ( x^{2}-2 \right )\left ( x^{2}+2 \right )\left ( \left ( x^{2}+2 \right )^{2}-\left ( 2x \right )^{2} \right )=\left ( x^{2}-2 \right )\left ( x^{2}+2 \right )\left ( x^{2}+2-2x \right )\left ( x^{2}+2+2x \right )=\left ( x^{2}-2 \right )\left ( x^{2}+2 \right )\left ( x^{2}-2x+2 \right )\left ( x^{2}+2x+2 \right )\)

Ответ: \(\left ( x^{2}-2 \right )\left ( x^{2}+2 \right )\left ( x^{2}-2x+2 \right )\left ( x^{2}+2x+2 \right )\)

Решение №17032: \(\sqrt{\left ( y^{2}+\frac{4}{y^{2}} \right )^{2}-8\left ( y+\frac{2}{y} \right )^{2}+48}=\sqrt{\left ( y+\frac{2}{y} \right )^{4}-8\left ( y+\frac{2}{y} \right )^{2}+16-8\left ( y+\frac{2}{y} \right )^{2}+48}=\sqrt{\left ( y+\frac{2}{y} \right )^{4}-16\left ( y+\frac{2}{y} \right )^{2}+64}=\sqrt{\left ( y^{2}+4+\frac{4}{y^{2}}-8 \right )^{2}}=\sqrt{\left ( \left ( y-\frac{2}{y} \right )^{2} \right )^{2}}=\left ( y-\frac{2}{y} \right )^{2}\)

Ответ: \(\left ( y-\frac{2}{y} \right )^{2}\)

Решение №17033: \(\left ( \frac{\left ( z^{\frac{2}{p}}+z^{\frac{2}{q}} \right )^{2}-4z^{\frac{2}{p}+\frac{2}{q}}}{\left ( z^{\frac{1}{p}}-z^{\frac{1}{q}} \right )^{2}+4z^{\frac{1}{p}+\frac{1}{q}}} \right )^{\frac{1}{2}}=\left ( \frac{\left ( z^{\frac{2}{p}}-z^{\frac{2}{q}} \right )}{\left ( z^{\frac{1}{p}}+z^{\frac{1}{q}} \right )^{2}} \right )^{\frac{1}{2}}=\left | z^{\frac{1}{p}}-z^{\frac{1}{q}} \right |\)

Ответ: \(\left | z^{\frac{1}{p}}-z^{\frac{1}{q}} \right |\)

Решение №17034: \(\sqrt[4]{32\sqrt[3]{4}}+\sqrt[4]{64\sqrt[3]{\frac{1}{2}}}-3\sqrt[3]{2\sqrt[4]{2}}=\sqrt[4]{2^{5}\cdot 2^{\frac{2}{3}}}+\sqrt[4]{2^{6}\cdot 2^{-\frac{1}{3}}}-3\sqrt[3]{2\cdot 2^{\frac{1}{4}}}=2^{\frac{17}{12}}+2^{\frac{17}{12}}-3\cdot 2^{\frac{5}{12}}=2\cdot 2^{\frac{17}{12}}-3\cdot 2^{\frac{5}{12}}=2^{\frac{5}{12}}\left ( 4-3 \right )=2^{\frac{5}{12}}=\sqrt[12]{32}\)

Ответ: \(\sqrt[12]{32}\)

Решение №17035: \(\sqrt[3]{\frac{2x^{2}}{9+18x+9x^{2}}}\sqrt{\frac{\left ( x+1 \right )\sqrt[3]{1-x}}{x}}\sqrt[3]{\frac{3\sqrt{1-x^{2}}}{2x\sqrt{x}}}=\sqrt[6]{\frac{4x^{4}}{81\left ( 1+x \right )^{4}}\cdot \frac{\left ( 1+x \right )^{3}\left ( 1-x \right )}{x^{3}}\frac{9\left ( 1-x^{2} \right )}{4x^{3}}}=\sqrt[6]{\frac{36x^{4}\left ( 1+x \right )^{4}\left ( 1-x \right )^{2}}{324x^{6}\left ( 1+x \right )^{4}}}=\sqrt[6]{\frac{\left ( 1-x \right )^{2}}{9x^{2}}}=\sqrt[3]{\frac{1-x}{3x}}\)

Ответ: \(\sqrt[3]{\frac{1-x}{3x}}\)

Решение №17036: \(\left ( \sqrt{\frac{\left ( 1-n \right )^{3}\sqrt{1+n}}{n}}\sqrt[3]{\frac{3n^{2}}{4-8n+4n^{2}}} \right )^{-1}:\sqrt[3]{\left ( \frac{3n\sqrt{n}}{2\sqrt{1-n^{2}}} \right )^{-1}}=\left ( \sqrt[6]{\left ( \frac{\left ( 1-n \right )^{3}\sqrt[3]{1+n}}{n^{3}} \right )^{3}}\sqrt[6]{\frac{3n^{2}}{4\left ( 1-2n+n^{2} \right )^{2}}} \right )^{-1}\sqrt[6]{\left ( \frac{3n\sqrt{n}}{2\sqrt{1-n^{2}}} \right )^{2}}=\left ( \sqrt[6]{\frac{\left ( 1-n \right )^{3}\left ( 1+n \right )9n^{4}}{n^{3}16\left ( 1-n \right )^{4}}} \right )^{-1}\sqrt[6]{\frac{9n^{3}}{4\left ( 1-n \right )\left ( 1+n \right )}}=\sqrt[6]{\frac{16\left ( 1-n \right )9n^{3}}{9n\left ( 1+n \right )4\left ( 1-n \right )\left ( 1+n \right )}}=\sqrt[6]{\frac{4n^{2}}{\left ( 1+n \right )^{2}}}=\sqrt[3]{\frac{2n}{1+n}}\)

Ответ: \(\sqrt[3]{\frac{2n}{1+n}}\)

Решение №17037: \(\sqrt[6]{6x\left (11+4\sqrt{6} \right )}\cdot \sqrt[3]{4\sqrt{2x}-2\sqrt{3x}}=\sqrt[6]{4x\left ( 11+4\sqrt{6} \right )}\cdot \sqrt[3]{2\sqrt{x}\left ( 2\sqrt{2}-\sqrt{3} \right )}=\sqrt[6]{4x\left (11+4\sqrt{6} \right )}\cdot \sqrt[6]{\left ( 2\sqrt{x}\left ( 2\sqrt{2}-\sqrt{3} \right ) \right )^{2}}=\sqrt[6]{4x\left ( 11+4\sqrt{6} \right )}\cdot \sqrt[6]{4x\left (11-4\sqrt{6} \right )}=\sqrt[6]{4x\left (11+4\sqrt{6} \right )4x\left (11-4\sqrt{6} \right )}=\sqrt[6]{16x^{2}\left ( 121-96 \right )}=\sqrt[6]{400x^{2}}=\sqrt[3]{20x}\)

Ответ: \(\sqrt[3]{20x}\)

Решение №17038: \(\frac{\sqrt[3]{\sqrt{3}+\sqrt{6}}\sqrt[6]{9-6\sqrt{2}}-\sqrt[6]{18}}{\sqrt[6]{2}-1}=-\sqrt[3]{3}; \frac{\sqrt[3]{\left ( \sqrt{3}+\sqrt{6} \right )^{2}}\sqrt[6]{9-6\sqrt{2}}-\sqrt[6]{18}}{\sqrt[6]{2}-1}=-\sqrt[3]{3};\frac{\sqrt[6]{9^{2}-\left ( 6\sqrt{2} \right )^{2}-\sqrt[6]{18}}}{\sqrt[6]{2}-1}=-\sqrt[3]{3}; \frac{\sqrt[6]{3^{2}}\left ( 1-\sqrt[6]{2} \right )}{\sqrt[6]{2}-1}=-\sqrt[3]{3};-\sqrt[3]{3}=-\sqrt[3]{3}\)

Ответ: \(-\sqrt[3]{3}=-\sqrt[3]{3}\)

Решение №17039: \(\left ( \frac{4-2x+x^{2}}{4-2x}+\frac{6x^{2}+8+12x}{4-x^{2}}-\frac{x^{2}+2x+4}{2x+4} \right )^{-\frac{1}{3}}\cdot \left ( x+2 \right )=\left ( -\frac{x^{2}-2x+4}{2\left ( x-2 \right )}-\frac{6x^{2}+12x+8}{\left ( x-2 \right )\left ( x+2 \right )}-\frac{x^{2}+2x+4}{2\left ( x+2 \right )} \right )^{-\frac{1}{3}}\cdot \left ( x+2 \right )=-\left ( \frac{2\left ( x^{3}+6x^{2}+12x+8 \right )}{2\left ( x-2 \right )\left ( x+2 \right )} \right )^{-\frac{1}{3}}\cdot \left ( x+2 \right )=-\sqrt[3]{\frac{\left ( x-2 \right )\left ( x+2 \right )^{3}}{\left ( x+2 \right )^{2}}}=-\sqrt[3]{x^{2}-4}=\sqrt[3]{4-x^{2}}\)

Ответ: \(\sqrt[3]{4-x^{2}}\)

Решение №17040: \(\frac{\sqrt[3]{ab}\left ( \sqrt[3]{b^{2}}-\sqrt[3]{a^{2}} \right )+\sqrt[3]{a^{4}}-\sqrt[3]{b^{4}}}{\sqrt[3]{a^{4}}+\sqrt[3]{a^{2}b^{2}}-\sqrt[3]{a^{3}b}}\cdot \sqrt[3]{a^{2}}=\frac{\left ( \sqrt[3]{a^{2}}-\sqrt[3]{b^{2}} \right )\left ( \sqrt[3]{a^{2}-\sqrt[3]{ab}+\sqrt[3]{b^{2}}} \right )}{\sqrt[3]{a^{2}}-\sqrt[3]{ab}+\sqrt[3]{b^{2}}}=\sqrt[3]{a^{2}}-\sqrt[3]{b^{2}}\)

Ответ: \(\sqrt[3]{a^{2}}-\sqrt[3]{b^{2}}\)

Решение №17041: \(\left ( \frac{\sqrt[3]{x+y}}{\sqrt[3]{x-y}}+\frac{\sqrt[3]{x-y}}{\sqrt[3]{x+y}}-2\right ):\left ( \frac{1}{\sqrt[3]{x-y}}-\frac{1}{\sqrt[3]{x+y}} \right )=\frac{\left ( \sqrt[3]{x+y} \right )^{2}-2\sqrt[3]{\left ( x+y \right )\left ( x-y \right )}+\left ( \sqrt[3]{x-y} \right )^{2}}{\sqrt[3]{x^{2}-y^{2}}}:\frac{\sqrt[3]{x+y}-\sqrt[3]{x-y}}{\sqrt[3]{x^{2}-y^{2}}}=\sqrt[3]{x+y}-\sqrt[3]{x-y}\)

Ответ: \(\sqrt[3]{x+y}-\sqrt[3]{x-y}\)

Решение №17042: \(\sqrt{\frac{\sqrt{\left ( a-y \right )\left ( y-b \right )}+\sqrt{\left ( a+y \right )\left ( y+b \right )}}{\sqrt{\left ( a+y \right )\left ( y+b \right )}-\sqrt{\left ( a-y \right )\left ( y-b \right )}}}=\sqrt{\frac{\left ( \sqrt{\left ( a-y \right )\left ( y-b \right )}+\sqrt{\left ( a+y \right )\left ( y+b \right )} \right )\left ( \sqrt{\left ( a+y \right )\left ( y+b \right )}+\sqrt{\left ( a-y \right )\left ( y-b \right )} \right )}{\left ( \sqrt{\left ( a+y \right )\left ( y+b \right )}\sqrt{\left ( a-y \right )\left ( y-b \right )} \right )\left ( \sqrt{\left ( a+y \right )\left ( y+b \right )}+\sqrt{\left ( a-y \right )\left ( y-b \right )} \right )}}=\sqrt{\frac{-y^{2}+\left ( a+b \right )y-ab+2\sqrt{-y^{4}+\left ( a^{2}+b^{2} \right )y^{2}-a^{2}b^{2}}}{y^{2}+\left ( a+b \right )y+ab+y^{2}-\left ( a+b \right )y+ab}}=\sqrt{\frac{\left ( a+b \right )y+\sqrt{-y^{4}+\left ( a^{2}+b^{2} \right )y^{2}-a^{2}b^{2}}}{2y^{2}+2ab}}=\sqrt{\frac{\left ( a+b \right )\sqrt{ab}+\sqrt{-a^{2}b^{2}+\left ( a^{2}+b^{2} \right )ab-a^{2}b^{2}}}{ab+ab}}=\sqrt{\frac{\left ( a+b \right )\sqrt{ab}+\sqrt{ab\left ( a^{2}-2ab+b^{2} \right )}}{2ab}}=\sqrt{\frac{a+b+\left | a-b \right |}{2\sqrt{ab}}}=\sqrt[4]{\frac{b}{a}};\sqrt[4]{\frac{a}{b}}\)

Ответ: \(\sqrt[4]{\frac{b}{a}};\sqrt[4]{\frac{a}{b}}\)

Решение №17043: \(\left ( \frac{\sqrt[4]{8}+2}{\sqrt[4]{2}+\sqrt[3]{2}}-\sqrt[3]{4} \right ):\left (\frac{\sqrt[4]{8}-2}{\sqrt[4]{2}-\sqrt[3]{2}} -3\sqrt[12]{128} \right )^{\frac{1}{2}}=\frac{\sqrt[12]{2^{9}}+\sqrt[12]{2^{12}}}{\sqrt[12]{2^{3}}+\sqrt[12]{2^{4}}}-\sqrt[12]{2^{8}}:\sqrt{\frac{\sqrt[12]{2^{9}}+\sqrt[12]{2^{12}}}{\sqrt[12]{2^{3}}+\sqrt[12]{2^{4}}}-3\sqrt[12]{2^{7}}}=\sqrt[12]{2^{6}}-\sqrt[12]{2^{7}}:\sqrt{\left ( \sqrt[12]{2^{6}}-\sqrt[12]{2^{7}} \right )^{2}}=\frac{\sqrt{2}\left ( 1-\sqrt[12]{2} \right )}{\sqrt[4]{2}\left ( \sqrt[12]{2}-1 \right )}=-\sqrt[4]{2}\)

Ответ: \(-\sqrt[4]{2}\)

Решение №17044: \(\left ( \frac{\sqrt[4]{x^{3}}-y}{\sqrt[4]{x}-\sqrt[3]{y}}-3\sqrt[12]{x^{3}y^{4}} \right )^{-\frac{1}{2}}\left ( \frac{\sqrt[4]{x^{3}}+y}{\sqrt[4]{x}+\sqrt[3]{y}}-\sqrt[3]{y^{2}} \right )=\left ( \sqrt[4]{x^{2}}-2\sqrt[4]{x\sqrt[3]{y}}+\sqrt[3]{y^{2}} \right )^{-\frac{1}{2}}\left ( \sqrt[4]{x^{2}}-\sqrt[4]{x\sqrt[3]{y}} \right )=-\sqrt[4]{x};\sqrt[4]{x}\)

Ответ: \(-\sqrt[4]{x};\sqrt[4]{x}\)

Решение №17045: \(\frac{\sqrt[3]{x+\sqrt{2-x^{2}}}\sqrt[6]{1-x\sqrt{2-x^{2}}}}{\sqrt[3]{1-x^{2}}}=\frac{\sqrt[6]{\left ( x+\sqrt{2-x^{2}} \right )^{2}}\sqrt[6]{1-x\sqrt{2-x^{2}}}}{\sqrt[3]{1-x^{2}}}=\frac{\sqrt[6]{\left ( 1+x\sqrt{2-x^{2}} \right )\left ( 1-x\sqrt{2-x^{2}} \right )}}{\sqrt[3]{1-x^{2}}}=\frac{\sqrt[6]{2\left ( 1-x^{2}\left ( 2-x^{2} \right )^{2} \right )}}{\sqrt[3]{1-x^{2}}}=\frac{\sqrt[6]{2\left ( 1-x^{2} \right )^{2}}}{\sqrt[3]{1-x^{2}}}=\frac{\sqrt[6]{2}\left | \sqrt[3]{1-x^{2}} \right |}{\sqrt[3]{1-x^{2}}}=-\sqrt[6]{2};\sqrt[6]{2}\)

Ответ: \(-\sqrt[6]{2};\sqrt[6]{2}\)

Решение №17046: \(\frac{\sqrt[3]{2a+2\sqrt{a^{2}-1}}}{\left ( \frac{\sqrt{a-1}}{\sqrt{a+1}}+\frac{\sqrt{a+1}}{\sqrt{a-1}}+2 \right )^{\frac{1}{3}}}=\frac{\sqrt[3]{\left ( \sqrt{a-1} \right )^{2}+2\sqrt{\left ( a-1 \right )\left ( a+1 \right )}+\left ( \sqrt{a+1} \right )^{2}}}{\sqrt[3]{\frac{\left ( \sqrt{a-1}+\sqrt{a+1} \right )^{2}}{\sqrt{a^{2}-1}}}}=\sqrt[3]{\left ( \sqrt{a-1}+\sqrt{a+1} \right )^{2}}\cdot \frac{\sqrt[6]{a^{2}-1}}{\sqrt[3]{\left ( \sqrt{a-1} +\sqrt{a+1}\right )^{2}}}=\sqrt[6]{a^{2}-1}\)

Ответ: \(\sqrt[6]{a^{2}-1}\)