A^6+b^6=?

a+b=10

a^2+b^2=60

Дано

{ a + b = 10

{ a^2 + b^2 = 60

Отсюда

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

100 = 60 + 2ab

ab = 20

(a^2 + b^2) ^2 = a^4 + 2a^2*b^2 + b^4 = a^4 + b^4 + 2 (ab) ^2

60^2 = a^4 + b^4 + 2*20^2

a^4 + b^4 = 60^2 - 2*20^2 = 3600 - 2*400 = 2800

Находим

a^6 + b^6 = (a^2) ^3 + (b^2) ^3 = (a^2 + b^2) (a^4 - a^2*b^2 + b^4) =

= 60 * (2800 - 20^2) = 60*2400 = 144000