已知x+1/x=3,求x的立方+1/x的立方的值,

问题描述:

已知x+1/x=3,求x的立方+1/x的立方的值,

∵x+1/x=3
∴x²+1/x²
=(x+1/x)²-2
=3²-2
=9-2
=7

∴x³+1/x³
= (x+1/x)(x²-1+1/x²)
=3*(7-1)
=3*6
=18

∵x+1/x=3
∴3³=27=(x+1/x)³=x³+3x²×1/x+3x×1/x²+1/x³=x³+1/x³+3x+3/x=x³+1/x³+3(x+1/x)=x³+1/x³+3×3
∴x³+1/x³=27-9=18

解;因为x+1/x=3
所以x²+1/x=(x+1/x)²-2
=3²-2
=9-2
=7
所以x³+1/x³=(x+1/x)(x²-1+1/x²)
=3×(7-1)
=3×6
=18

直接x+1/x=3三次方
展开得X^3+(1/X)^3+3(x+1/x)=27
X^3+(1/X)^3=27-9=18

x+1/x=3
﹙x+1/x﹚²=3²
x²+2+1/x²=9
x²+1/x²=7
∴x³+1/x³=﹙x+1/x﹚﹙x²-1+1/x²﹚
=3×﹙7-1﹚
=18