4/

`(a^2+b^2)(a^4+b^4)≤(a+b)(a^5+b^5)`

`⇔a^6+a^4b^2+a^2b^4+b^6≤a^6+ab^5+a^5b+b^6`

`⇔a^2b^2(a^2+b^2)≤ab(a^4+b^4)`

`⇔a^4+b^4-ab(a^2+b^2)≥0`

`⇔4a^4+4b^4-4ab(a^2+b^2)≥0`

`⇔a^4+2a^2b^2+b^4-4ab(a^2+b^2)+4a^2b^2+3a^4+3b^4-6a^2b^2≥0`

`⇔(a^2+b^2)^2-4ab(a^2+b^2)+4a^2b^2+3(a^4+b^4-2a^2b^2)≥0`

`⇔(a^2+b^2-2ab)^2+3(a^2-b^2)^2≥0`

`⇔(a+b)^4+3(a^2-b^2)^2≥0` (luôn đúng)

`⇒đpcm`

5/

`(a^10+b^10)(a^2+b^2)≥(a^8+b^8)(a^4+b^4)`

`⇔a^12+a^2b^10+a^10b^2+b^12≥a^12+a^4b^8+a^8b^4+b^12`

`⇔a^2b^2(a^8+b^8)≥a^4b^4(a^4+b^4)`

`⇔a^8+b^8-a^2b^2(a^4+b^4)≥0`

`⇔4a^8+4b^8-4a^2b^2(a^4+b^4)≥0`

`⇔a^8+2a^4b^4+b^8-4a^2b^2(a^4+b^4)+4a^4b^4+3a^8+3b^8-6a^4b^4`

`⇔(a^4+b^4)^2-4a^2b^2(a^4+b^4)+4a^4b^4+3(a^8+b^8-2a^4b^4)≥0`

`⇔(a^4+b^4-2a^2b^2)^2+3(a^4-b^4)^2≥0`

`⇔(a^2+b^2)^4+3(a^4+b^4)^2≥0` (luôn đúng)

`⇒đpcm`

6/

`(a^2+b^2)(a^3+b^3)≤(a+b)(a^4+b^4)`

`⇔a^5+b^5+a^2b^3+a^3b^2+b^5≤a^5+a^4b+ab^4+b^5`

`⇔a^2b^2(a+b)≤ab(a^3+b^3)`

`⇔a^3+b^3≥ab(a+b)`

`⇔(a+b)(a^2-ab+b^2)-ab(a+b)≥0`

`⇔(a+b)(a^2-2ab+b^2)≥0`

`⇔(a+b)(a-b)^2≥0`

Bài này có thêm `a,b` dương thì mới làm đc nhé