TITLE: Prove the inequality is true
QUESTION [12 upvotes]: Here is a question that I need to prove 

Prove that for $a, b \geq 0$
$$a^8+b^8\geq a^3b^5+a^5b^3$$

So far I have managed to simplify to
$$(a^3-b^3)(a^5-b^5)\geq 0$$

REPLY [2 votes]: WLOG, consider $a\ge b$. So:
$$a^3\ge b^3;a^5\ge b^5$$
Use Rearrangement inequality:
$$a^8+b^8\ge a^3b^5+b^3a^5.$$