Question

先化简，在求值：(x+根号xy)/(y+根号xy)+((根号xy)-y)/(x-根号xy),其中x=根号3+1,y=根号3-1

先化简，再求值：(x+根号xy)/(y+根号xy)+((根号xy)-y)/(x-根号xy),其中x=根号3+1,y=根号3-1

Answer 1

解：(x+根号xy)/(y+根号xy)+((根号xy)-y)/(x-根号xy),
=√x(√x+√y)/√y(√x+√y)+√y(√x-√y)/√x(√x-√y)
=√x/√y+√y/√x=(x+y)/√xy
x=√3+1 y=√3-1 ∴x+y=2√3 xy=2
∴(x+y)/√xy=2√3/2=√3

Answer 2

[x+√(xy)]/[y+√(xy)]+[(√(xy)-y]/[x-√(xy)]
=√x(√x+√y)/[√y(√y+√x)]+√y(√x-√y)/[√x(√x-√y)]
=√x/√y+√y/√x
=x/√(xy)+y/√(xy)
=(x+y)/√(xy)
=(√3+1+√3-1)/√[(√3+1)(√3-1)]
=2√3/2
=√3

Answer 3

解：（x＋√xy）/（y＋√xy）+（√xy-y）/（x-√xy）
=[（x＋√xy）×（x-√xy）+（√xy-y）×（y＋√xy）]/[（y＋√xy）×（x-√xy）]
=（x²-xy+xy-y²）/（xy+x√xy-y√xy-xy）
=（x²-y²）/[（x-y）√xy]
=（x-y）（x+y）/[（x-y）√xy]
=（x+y）/√xy
=（√3+1+√3-1）/√（√3+1）（√3-1）
=2√3/√2
=√6