Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 477

\[\boxed{\mathbf{477}.}\]

\[1)\ a - x = a^{1} - x^{1} = a^{\frac{3}{3}} - x^{\frac{3}{3}} =\]

\[= \left( a^{\frac{1}{3}} \right)^{3} - \left( x^{\frac{1}{3}} \right)^{3} =\]

\[= \left( a^{\frac{1}{3}} - x^{\frac{1}{3}} \right) \bullet\]

\[\bullet \left( \left( a^{\frac{1}{3}} \right)^{2} + a^{\frac{1}{3}}x^{\frac{1}{3}} + \left( x^{\frac{1}{3}} \right)^{2} \right) =\]

\[= \left( a^{\frac{1}{3}} - x^{\frac{1}{3}} \right)\left( a^{\frac{2}{3}} + a^{\frac{1}{3}}x^{\frac{1}{3}} + x^{\frac{2}{3}} \right);\]

\[2)\ x^{\frac{3}{2}} - y^{\frac{3}{2}} = \left( x^{\frac{1}{2}} \right)^{3} - \left( y^{\frac{1}{2}} \right)^{3} =\]

\[= \left( x^{\frac{1}{2}} - y^{\frac{1}{2}} \right) \bullet\]

\[\bullet \left( \left( x^{\frac{1}{2}} \right)^{2} + x^{\frac{1}{2}}y^{\frac{1}{2}} + \left( y^{\frac{1}{2}} \right)^{2} \right) =\]

\[= \left( x^{\frac{1}{2}} - y^{\frac{1}{2}} \right)\left( x + x^{\frac{1}{2}}y^{\frac{1}{2}} + y \right);\]

\[3)\ a^{\frac{1}{2}} - b^{\frac{1}{2}} = a^{\frac{3}{6}} - b^{\frac{3}{6}} =\]

\[= \left( a^{\frac{1}{6}} \right)^{3} - \left( b^{\frac{1}{6}} \right)^{3} =\]

\[= \left( a^{\frac{1}{6}} - b^{\frac{1}{6}} \right) \bullet\]

\[\bullet \left( \left( a^{\frac{1}{6}} \right)^{2} + a^{\frac{1}{6}}b^{\frac{1}{6}} + \left( b^{\frac{1}{6}} \right)^{2} \right) =\]

\[= \left( a^{\frac{1}{6}} - b^{\frac{1}{6}} \right)\left( a^{\frac{1}{3}} + a^{\frac{1}{6}}b^{\frac{1}{6}} + b^{\frac{1}{3}} \right);\]

\[4)\ 27a + c^{\frac{1}{2}} = 3^{3}a^{\frac{3}{3}} + c^{\frac{3}{6}} =\]

\[= \left( 3a^{\frac{1}{3}} \right)^{3} + \left( c^{\frac{1}{6}} \right)^{3} =\]

\[= \left( 3a^{\frac{1}{3}} + c^{\frac{1}{6}} \right) \bullet\]

\[\bullet \left( \left( 3a^{\frac{1}{3}} \right)^{2} - 3a^{\frac{1}{3}}c^{\frac{1}{6}} + \left( c^{\frac{1}{6}} \right)^{2} \right) =\]

\[= \left( 3a^{\frac{1}{3}} + c^{\frac{1}{6}} \right) \bullet\]

\[\bullet \left( 9a^{\frac{2}{3}} - 3a^{\frac{1}{3}}c^{\frac{1}{6}} + c^{\frac{1}{3}} \right).\]