Bài `1`
`a+b=25⇒(a+b)^2=25^2=625`
`ab=136 ⇒ 2ab = 136.2 = 272`
`⇒a^2+b^2 = (a+b)^2-2ab = 625-272=353.`
`a^3+b^3=(a+b)(a^2-ab+b^2)=25.(353 -136)=217.25=5425.`
Ta có: `a^4+a^3b+ab^3+b^4=(a^4+b^4)+a^2b^2(a+b)`
`=a^3.(a+b) +b^3(a+b)`
`=(a+b)(a^3+b^3)`
`⇒a^4+b^4 =(a+b)(a^3+b^3) - a^2b^2(a+b)= 136. 5425 - 136^2 . 25= 737800 - 462400 = 275400.`
`b)`
`x+y=9⇒(x+y)^2=9^2=81`
`xy=18 ⇒ 2xy= 36`
`⇒M=x^2+y^2 = (x+y)^2-2xy = 81-36=45.`
`x^3+y^3=(x+y)(x^2-xy+y^2)=9.(45-18)=9. 27 =243`
Ta có: `x^4+x^3y+xy^3+y^4=(x^4+y^4)+x^2y^2(x+y)`
`=x^3.(x+y) +y^3(x+y)`
`=(x+y)(x^3+y^3)`
`⇒N=x^4+y^4 =(x+y)(x^3+y^3) - x^2y^2(x+y)=18. 243 - 18^2 . 9 = 1458.`
`P= x^2-y^2 `
Có `x^2 + y^2 = 45`
`⇒ x^2 -2xy + y^2 = 45- 2xy`
`⇔(x-y)^2 = 45- 2.18 = 45- 36 =9`
`⇒x-y=±3`
Xét hai trường hợp:
`x^2-y^2= (x+y)(x-y) = 9. 3 = 27`
`x^2-y^2= (x+y)(x-y) = 9. (-3) = -27.`
`Q=x^3 - y^3=(x-y)(x^2+xy+y^2)= 3. (45+18)= 3. 63 = 189.`
`Q=x^3 - y^3=(x-y)(x^2+xy+y^2)= -3. (45+18)= -3. 63 = -189.`
