Question

已知x+1/x=3,求x^3/2-x^-3/2的值

Answer 1

俊狼猎英团队为您解答

立方和 a³+b³=(a+b)(a²-ab+b²)
立方差 a³-b³=(a-b)(a²+ab+b²)

x^3/2-x^-3/2
=(X^1/2-X^-1/2)(X+1+1/X)
=4(X^1/2-X^-1/2)

(X^1/2-X^-1/2)^2=X+1/X-2=1
∴X^1/2-X^(-1/2)=±1，
∴原式=±4。

Answer 2

因：x+1/x=3
所以有：(x+1/x)^2=9 展开整理得：
x^2+1/x^2=7
(x-1/x)^2=x^2-2+1/x^2=5
所以：x-1/x=±√

x^3/2-x^(-3)/2
=1/2(x-1/x)(x^2+1+1/x^2)
=1/2(±√5)(7+1)
=±4√5

追问

不好意思，那个是求x的3/2次方-x的-3/2次方的值

追答

因：[x^(1/2)-x^(-1/2)]^2=x-2+1/x=3-2=1
所以可得：x^(1/2)-x^(-1/2)=±1

x^(3/2)-x^(-3/2)
=[x^(1/2)-x^(-1/2)](x+1+1/x)
=±1x4
=±4