Question 763882
Radii of the three balls before melting are 5, 4, and 3.  Their summed volume is {{{(4/3)*pi*5^3+(4/3)*pi*4^3+(4/3)*pi*3^3=highlight((4/3)*pi*(5^3+4^3+3^3))}}}


All of that material made into a new single bigger ball will be that same volume.  You just need to use ONE variable for radius, or diameter.  
Call this radius, r.


Equate the volumes.
{{{(4/3)*pi*(5^3+4^3+3^3)=(4/3)*pi*r^3}}}


You can simplify very easily first:
{{{cross((4/3)*pi)*(5^3+4^3+3^3)=cross((4/3)*pi)*r^3}}}
{{{highlight(r^3=5^3+4^3+3^3)}}}
Find or evaluate r.