已知x+1/x=5/2,求x^3+1/(x^3)

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已知x+1/x=5/2,求x^3+1/(x^3)

已知x+1/x=5/2,求x^3+1/(x^3)
已知x+1/x=5/2,求x^3+1/(x^3)

已知x+1/x=5/2,求x^3+1/(x^3)

x+1/x=5/2
两边平方得:
x²+2+1/x²=25/4

x²+1/x²=25/4-2=17/4
∴x³+(1/x)³
=(x+1/x)(x²-1+1/x²)
=5/2×(17/4-1)
=5/2×(13/4)
=65/8

x²+1/x²

=(x+1/x)²-2
=(5/2)²-2
=17/4

于是
x³+1/x³

=(x+1/x)(x²-x*1/x+1/x²)

=(x+1/x)(x²+1/x²-1)

=5/2*(17/4-1)

=5/2*13/4

=65/8


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