A square root problem

[littlemathquark]

Homework Statement

$\sqrt{\dfrac{3^8+5^8+34^4}2} =?$

Relevant Equations

None

I can't find a short solution without using calculator.

 

[Halc]

Consider the prime factors of all the components
Oh, that only works if the 3 upper terms are multiplied.
Adding them makes only for a big unwieldy number. Don't see how to simplify it at all.

 

[Ibix]

Do 3, 5, and 34 (or their prime factors) have any properties in common that might help? Especially helpful if there are terms that might cancel when expanded.

 

[littlemathquark]

Do 3, 5, and 34 (or their prime factors) have any properties in common that might help? Especially helpful if there are terms that might cancel when expanded.

$3^8+5^8+34^4=9^4+25^4+34^4=9^4+25^4+(9+25)^4$
That's all and I'm stuck.

 

[Ibix]

Interesting! Not what I had in mind at all, but it also works.

What do you get if you expand that? I found it helpful to write $x=3$ and $y=5$ so I didn't lose track of my special numbers among all the other ones.

If the expansion is going to help you in this particular problem, what has to be true about it?