化简无理指数幂.主要是思路啊大侠们遇到这种题如何插手啊化简（X+Y)/(X^(1/3)+Y^(1/3))-(X^(4/3)-Y^(4/3))/(X^(2/3)-Y^(2/3))咋做啊

Question

化简无理指数幂.主要是思路啊大侠们 遇到这种题如何插手啊
化简 （X+Y)/(X^(1/3)+Y^(1/3)) - (X^(4/3)-Y^(4/3))/(X^(2/3)-Y^(2/3)) 咋做啊

Answer 1

第一个乘X^(1/3)-Y^(1/3)就行了- -
(X+Y)/(X^(1/3)+Y^(1/3)) - (X^(4/3)-Y^(4/3))/(X^(2/3)-Y^(2/3))
= (X+Y)(X^(1/3)-Y^(1/3))/(X^(1/3)+Y^(1/3))(X^(1/3)-Y^(1/3)) - (X^(4/3)-Y^(4/3))/(X^(2/3)-Y^(2/3))
= (X+Y)(X^(1/3)-Y^(1/3))/(X^(2/3)-Y^(2/3)) - (X^(4/3)-Y^(4/3))/(X^(2/3)-Y^(2/3))
= [(X+Y)(X^(1/3)-Y^(1/3)) - (X^(4/3)-Y^(4/3))]/(X^(2/3)-Y^(2/3))
= [(X^(4/3)-XY^(1/3) + X^(1/3)Y-Y^(4/3) - X^(4/3) + Y^(4/3)]/(X^(2/3)-Y^(2/3))
= (X^(1/3)Y - XY^(1/3))/(X^(2/3)-Y^(2/3))
= (X^(1/3)Y - XY^(1/3))(X^(4/3)+X^(2/3)Y^(2/3)+Y^(4/3))/(X^(2/3)-Y^(2/3))(X^(4/3)+X^(2/3)Y^(2/3)+Y^(4/3))
= (X^(1/3)Y - XY^(1/3))(X^(4/3)+X^(2/3)Y^(2/3)+Y^(4/3))/(X^2-Y^2)
= (X^(5/3)Y + XY^(5/3) + X^(1/3)Y^(7/3) - X^(7/3)Y^(1/3) - X^(5/3)Y - XY^(5/3))/(X^2-Y^2)