Calculator at rescue?

Note that $147+219=366$ . If we say $a=147$ , $b=219$ , we want to find $\begin{aligned} \sqrt{\frac{a^4+b^4+(a+b)^4}{2}} &= \sqrt{\frac{a^4+b^4+a^4+4a^3 b+6a^2 b^2+4ab^3+b^4}{2}} \\ &=\sqrt{a^4+2a^3 b+3a^2 b^2+2ab^3+b^4} \\ &=a^2+ab+b^2 \end{aligned}$

I'd still need a calculator here, but at least it's simpler!

In case the square-rooting step looks like it's come out of nowhere, read about trinomial coefficients .