考點(diǎn):有理數(shù)指數(shù)冪的運(yùn)算性質(zhì)
專題:函數(shù)的性質(zhì)及應(yīng)用
分析:由已知得x2-y2=2(y-x),所以x=y或x+y=-2.由此分類討論,能求出結(jié)果.
解答:
解:∵x
2=2y+5,y
2=2x+5(x≠y),
∴x
2-y
2=2(y-x),即(x-y)(x+y)=2(y-x)
∴x=y或x+y=-2.
當(dāng)x=y時(shí),x
2=2x+5,解得x=1
±,①
x
3-2x
2y
2+y
3=2x
2(x-x
2)=2(2x+5)(x-2x-5)
=-2(2x
2+15x+25)
=-38x-70
=-108
±38,
當(dāng)x+y=-2時(shí),x,y是方程x
2+2x-1=0兩根,
則x+y=-2,且xy=-1,
x
3-2x
2y
2+y
3=(x+y)[(x+y)
2-3xy]-2(xy)
2=-16,
綜上,x
3-2x
2y
2+y
3的值為-108
±38或-16.
故答案為:-108
±38或-16.
點(diǎn)評(píng):本題考查有理數(shù)指數(shù)冪的化簡(jiǎn)求值,是中檔題,解題時(shí)要注意分類討論思想的合理運(yùn)用.