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

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 .