已知x^1/2+x^-1/2=3,求x^3/2+x^-3/2+2(分子)/x+x^-1 +3(分母)的值

题目

已知x^1/2+x^-1/2=3,求x^3/2+x^-3/2+2(分子)/x+x^-1 +3(分母)的值

答案

x^1/2+x^(-1/2)=3 …………（1）
1、（1）式两边取平方,即（x^1/2+x^(-1/2)）^2 = 3^2,化简得
x + x^(-1) + 2 = 9
即x + x^(-1) = 7 ………………（2）
2、（1）式两边取立方,即（x^1/2+x^(-1/2)）^3 = 3^3,化简得
x^3/2 + x^(-3/2) + 3x *x^(-1/2) + 3x^(1/2) *x^(-1)
= x^3/2 + x^(-3/2) + 3x^(1/2) + 3x^(-1/2)
= x^3/2 + x^(-3/2) + 3(x^(1/2) + x^(-1/2))
= x^3/2 + x^(-3/2) + 9 = 3^3 = 27
所以x^3/2 + x^(-3/2) = 18
3、（2）式两边取平方,即（x + x^(-1)）^2 = 7^2,化简得
x^2+x^(-2) + 2 = 49
即x^2+x^(-2) = 47
[x^3/2+x^(-3/2)+2]/[x+x^(-1)+3]
= (18 +2) / (7+3)
= 2