已知x+1/x=7 求x^(3/2)+x^(-3/2)

x+1/x=7,
x^2+1/x^2=7^2-2=47,
7*47=(x+1/x)*(x+1/x^2)=x^3+1/x^3+x+1/x=x^3+1/x^3+7
则x^3+1/x^3=7*47-7=322
所以(x^(3/2)+x^(-3/2))^2=x^3+1/x^3+2=322+2=324
x^(3/2)+x^(-3/2)=324^(1/2)=18

x+1/x=7
x^2 - 7x + 1 = 0
从求根公式容易判断 x>0 。因此 可对x 开平方。

设 y = √x
那么原题目化为
y^2 + 1/y^2 = 7，求 y^3 + 1/y^3 的值。

由 y^2 + 1/y^2 = 7 得到

(y + 1/y)^2 = 9

而
y^3 + 1/y^3
= (y + 1/y)*(y^2 - 1 + 1/y^2)
= √9 *（y^2 + 1/y^2 -1）
=3 * ( 7 - 1)
= 18

x+1/x = [x^(1/2)+x^(-1/2)]^2 -2 = 7
所以 x^(1/2)+x^(-1/2) = 3
根据 a^3+b^3 = (a+b)(a^2-ab+b^2)
所以
x^(3/2)+x^(-3/2)
=[x^(1/2)+x^(-1/2)](x-1+1/x)
=3*(7-1)
=18

是x+1/x=7, x²+1/x²=7²-2=47, 7*47=(x+1/x)(x+1/x²=x³+1/x³+x+1/x=x³+1/x³+7
则x³+1/x³=7*47-7=322
所以(x^(3/2)+x^(-3/2))^2=x^3+1/x³+2=322+2=324
x^(3/2)+x^(-3/2)=324^(1/2)=18
个破题，讨厌死了

是x+1/x=