<s>[INST] Prove the inequality is true 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
$$(a^3-b^3)(a^5-b^5)\geq 0$$
 [/INST] I don't think an expansion is needed at all. Here is an even more trivial proof.
Consider that if $a>b$,  $a^3 > b^3$ and $a^5> b^5$ since $a$ and $b$ are positive. 
So, we have that $$(a^3-b^3)(a^5-b^5) > 0$$ when $a>b\geq 0 \tag{1}$.
Now, if $a\leq b$, 
$b^5\geq a^5$ and $b^3 \geq a^3$
so we have that $$(b^3-a^3)(b^5-a^5) = (a^3-b^3)(a^5-b^5) \geq 0 \tag{2}$$ when $b\geq a\geq 0$. 
$(1)$ and $(2)$ are together necessary and sufficient condition to prove the required inequality, i.e. that $$\boxed{a^8+b^8 \geq a^3b^5 + a^5b^3} 
~~~\blacksquare$$
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