题目

(1)已知x=1/2+根号3,xy=1,则x^2y-xy^2/x^2-y^2=_____.
(2)化简1+2根号3+根号5/(1+根号3)(根号3+根号5)=______.
(3)已知x=根号3+1,y=根号3-1,则x^2-2xy+y^2/x^2-y^2=______.
(4)已知a+1/a=5,则a^4+a^2+1/a^2=_____.

提问时间：2020-07-24

答案

(1)已知x=1/2+根号3=2-√3,xy=1,
y=2+√3;
则x^2y-xy^2/x^2-y^2
=xy(x-y)/(x+y)(x-y)
=xy/(x+y)
=1/(2+√3+2-√3)
=1/4;
(2)化简1+2根号3+根号5/(1+根号3)(根号3+根号5)
=(1+2√3+√5)(√3-1)/(√5+√3)(√3+1)(√3-1)
=(√3-1+6-2√3+√15-√5)/2(√5+√3)
=(√15+5-√5-√3)(√5-√3)/2（√5+√3）(√5-√3)
=(5√3-3√5+5√5-5√3-5+√15-√15+3)/4;
=(2√5+3)/4;
(3)已知x=根号3+1,y=根号3-1,则x^2-2xy+y^2/x^2-y^2
=(x-y)²/(x+y)(x-y)
=(x-y)/(x+y)
=(√3+1-√3+1)/(√3+1+√3-1)
=2/2√3
=√3/3;
(4)已知a+1/a=5,则a^4+a^2+1/a^2
=a²+1+1/a²
=(a+1/a)²-1;
=5²-1
=24;
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