一道有关分数指数幂的数学题已知x+x^(-1)=4,求下列各式的值（1）x^(1/2)+x^(-1/2)(2) x^(3/2)+x^(-3/2)(3) x^(1/2)-x^(-1/2)

Question

一道有关分数指数幂的数学题
已知x+x^(-1)=4,求下列各式的值
（1）x^(1/2)+x^(-1/2)
(2) x^(3/2)+x^(-3/2)
(3) x^(1/2)-x^(-1/2)

Answer 1

已知x+x^(-1)=4,两边平方：x²+2+x^(-2)=16,得x²+x^(-2)=14
（1） [x^(1/2)+x^(-1/2)]²=x+2+x^(-1)=4+2=6,
故x^(1/2)+x^(-1/2)=√6 .
(2) [x^(3/2)+x^(-3/2)]²=x^3+x^(-3)+2=[x+x^(-1)][x²-1+x^(-2)]+2
=4×(14-1)+2=50,
故x^(3/2)+x^(-3/2)=√50 .
(3) [x^(1/2)-x^(-1/2)]²= x-2+x^(-1)=4-2=2,
故x^(1/2)-x^(-1/2)=±√2 .