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已知x+y=1,x3+y3=三分之一,求x的五次方+y的五次方的值,最好通俗易懂
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已知x+y=1,x3+y3=三分之一,求x的五次方+y的五次方的值,最好通俗易懂
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答案和解析
x+y=1,x3+y3=三分之一
(x+y)(x²-xy+y²)=1/3
(x+y)³-3xy(x+y)=1/3
1-3xy=1/3
3xy=2/3
xy=2/9
x²+y²=1-2xy=1-4/9=5/9
所以
x的五次方+y的五次方
=(x²+y²)(x³+y³)-x²y³-x³y²
=5/9*1/3-x²y²(x+y)
=5/27-(2/9)²
=5/27-4/81
=11/81
(x+y)(x²-xy+y²)=1/3
(x+y)³-3xy(x+y)=1/3
1-3xy=1/3
3xy=2/3
xy=2/9
x²+y²=1-2xy=1-4/9=5/9
所以
x的五次方+y的五次方
=(x²+y²)(x³+y³)-x²y³-x³y²
=5/9*1/3-x²y²(x+y)
=5/27-(2/9)²
=5/27-4/81
=11/81
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