Страница 34 номер 89, ГДЗ по алгебре за 10-11 класс Алимов - ответы и решение
Упростить выражение (87–89). 1) \(\frac{x + y}{x^{\frac{2}{3}} - x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}}} + \frac{x - y}{x^{\frac{2}{3}} + x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}}} - \frac{x^{\frac{2}{3}} - y^{\frac{2}{3}}}{x^{\frac{1}{3}} - y^{\frac{1}{3}}}\); 2) \(\frac{(a - b)^2}{a^{\frac{3}{2}} - b^{\frac{3}{2}}} + \frac{a^2 - b^2}{\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right)\left(a + a^{\frac{1}{2}}b^{\frac{1}{2}} + b\right)}\); 3) \(\left(\frac{3x^{\frac{2}{3}} + 5x^{\frac{1}{3}}}{x + 1} + \frac{1}{x^{\frac{1}{3}} + 1}\right) : \left(4x^{\frac{1}{3}} + 4 + \frac{1}{x^{\frac{1}{3}}}\right)\).
1) \(\frac{x + y}{x^{\frac{2}{3}} - x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}}} + \frac{x - y}{x^{\frac{2}{3}} + x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}}} - \frac{x^{\frac{2}{3}} - y^{\frac{2}{3}}}{x^{\frac{1}{3}} - y^{\frac{1}{3}}} = \frac{(x + y)\left(x^{\frac{1}{3}} + y^{\frac{1}{3}}\right)}{\left(x^{\frac{1}{3}} + y^{\frac{1}{3}}\right)\left(x^{\frac{2}{3}} - x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}}\right)} +\)
\(+ \dfrac{(x - y)\left(x^{\frac{1}{3}} - y^{\frac{1}{3}}\right)}{\left(x^{\frac{1}{3}} - y^{\frac{1}{3}}\right)\left(x^{\frac{2}{3}} + x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}}\right)} - \dfrac{x^{\frac{2}{3}} - y^{\frac{2}{3}}}{x^{\frac{1}{3}} - y^{\frac{1}{3}}} =\)
\(= x^{\frac{1}{3}} + y^{\frac{1}{3}} + x^{\frac{1}{3}} - y^{\frac{1}{3}} - \dfrac{\left(x^{\frac{1}{3}} - y^{\frac{1}{3}}\right)\left(x^{\frac{1}{3}} + y^{\frac{1}{3}}\right)}{x^{\frac{1}{3}} - y^{\frac{1}{3}}} =\)
\(= 2x^{\frac{1}{3}} - x^{\frac{1}{3}} - y^{\frac{1}{3}} = x^{\frac{1}{3}} - y^{\frac{1}{3}};\)
2) \(\frac{(a - b)^{2}}{a^{\frac{3}{2}} - b^{\frac{3}{2}}} + \frac{a^{2} - b^{2}}{\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right)\left(a + a^{\frac{1}{2}}b^{\frac{1}{2}} + b\right)} =\)
\(= \frac{(a - b)^{2}}{a^{\frac{3}{2}} - b^{\frac{3}{2}}} + \frac{(a + b)\left(a^{\frac{1}{2}} - b^{\frac{1}{2}}\right)\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right)\left(a^{\frac{1}{2}} - b^{\frac{1}{2}}\right)}{\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right)\left(a + a^{\frac{1}{2}}b^{\frac{1}{2}} + b\right)\left(a^{\frac{1}{2}} - b^{\frac{1}{2}}\right)}\)
\(= \dfrac{(a - b)^{2}}{a^{\frac{3}{2}} - b^{\frac{3}{2}}} + \dfrac{(a + b)\left(a^{\frac{1}{2}} - b^{\frac{1}{2}}\right)^{2}}{a^{\frac{3}{2}} - b^{\frac{3}{2}}} =\)
\(= \dfrac{a^{2} - 2ab + b^{2} + (a + b)(a - 2a^{\frac{1}{2}}b^{\frac{1}{2}} + b)}{a^{\frac{3}{2}} - b^{\frac{3}{2}}} =\)
\(= \dfrac{a^{2} - 2ab + b^{2} + a^{2} - 2a^{\frac{3}{2}}b^{\frac{1}{2}} + ab + ab - 2a^{\frac{1}{2}}b^{\frac{3}{2}} + b^{2}}{a^{\frac{3}{2}} - b^{\frac{3}{2}}} =\)
\(= \dfrac{2a^{2} + 2b^{2} - 2a^{\frac{3}{2}}b^{\frac{1}{2}} - 2a^{\frac{1}{2}}b^{\frac{3}{2}}}{a^{\frac{3}{2}} - b^{\frac{3}{2}}} =\)
\(\dfrac{2a^{\frac{1}{2}}\left(a^{\frac{3}{2}} - b^{\frac{3}{2}}\right) - 2b^{\frac{1}{2}}\left(a^{\frac{3}{2}} - b^{\frac{3}{2}}\right)}{a^{\frac{3}{2}} - b^{\frac{3}{2}}} = 2\left(a^{\frac{1}{2}} - b^{\frac{1}{2}}\right);\)
3) \(\left(\frac{3x^{\frac{2}{3}} + 5x^{\frac{1}{3}}}{x + 1} + \frac{1}{x^{\frac{1}{3}} + 1}\right) : \left(4x^{\frac{1}{3}} + 4 + \frac{1}{x^{\frac{1}{3}}}\right) = \left(\frac{3x^{\frac{2}{3}} + 5x^{\frac{1}{3}}}{x + 1} + \frac{x^{\frac{2}{3}} - x^{\frac{1}{3}} + 1}{x + 1}\right) :\)
\(: \left(\dfrac{1}{x^{\frac{1}{3}}}\left(4x^{\frac{2}{3}} + 4x^{\frac{1}{3}} + 1\right)\right) =\)
\(= \left(\dfrac{4x^{\frac{2}{3}} + 4x^{\frac{1}{3}} + 1}{x + 1}\right) : \left(\dfrac{1}{x^{\frac{1}{3}}}\left(4x^{\frac{2}{3}} + 4x^{\frac{1}{3}} + 1\right)\right) =\)
\(= \dfrac{x^{\frac{1}{3}}}{x + 1};\)
