<s>[INST] Is there a trick to prove that $(x^4+y^4)^{1/2} \leq x^2+y^2$? $(x^{4}+y^{4})^{1/2} \leq  x^{2}+y^{2}$
I tried multiplying the original by 1 = $\displaystyle \frac{(x^{4}+y^{4})^{1/2}}{(x^{4}+y^{4})^{1/2}}$, but that just brings me back to the original.
Ive looked up sqrt and exponent laws and can't find anything helpful. What am I missing?
 [/INST] $$
\begin{align}
(x^2 + y^2)^2 &= x^4 + 2x^2y^2 + y^4\\
&= (x^4 + y^4) + 2x^2y^2
\end{align}
$$
Now compare $(x^4 + y^4)^{\frac{1}{2}}$ with $\left[(x^4 + y^4) + 2x^2y^2\right]^{\frac{1}{2}}$, knowing that for all $x, y$, $2x^2y^2\geq 0$.
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