已知a+b=3,ab=1,求a^7+b^7的值.
4个回答

a^3+b^3

=(a+b)(a^2-ab+b^2)

=(a+b)(a^2+2ab+b^2-3ab)

=(a+b)[(a+b)^2-3ab]

=3*(3^2-3*1)

=18

a^4+b^4

=(a^2+b^2)^2-2a^2b^2

=[(a^2+b^2+2ab)-2ab]^2-2(ab)^2

=[(a+b)^2-2ab]^2--2(ab)^2

=[3^2-2*1]^2-2*1^2

=49-2

=47

a^7+b^7

=(a^3+b^3)*(a^4+b^4)-(a^3b^4+a^4b^3)

=(a^3+b^3)*(a^4+b^4)-a^3b^3(a+b)

= (a^3+b^3)*(a^4+b^4)-(ab)^3(a+b)

=18*47-1^3*3

=846-3

=843