[tex]\displaystyle\bf\\U\left(2^{2013}+5^{565}+3^{122}\right)=\\\\=U\left(2^{2012+1}+5^{565}+3^{120+2}\right)=\\\\=U\left(2^{4\times503+1}+5^{565}+3^{4\times30+2}\right)=\\\\=U\left(\Big(2^4\Big)^{503}\times2+5^{565}+\Big(3^4\Big)^{30}\times3^2\right)=[/tex]
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[tex]\displaystyle\bf\\=U\left(U\Big(16^{503}\times2\Big)+U\Big(5^{565}\Big)+U\Big(81^{30}\times3^2\Big)\right)=\\\\\\=U\left(U\Big(6^{503}\times2\Big)+U\Big(5^{565}\Big)+U\Big(1^{30}\times3^2\Big)\right)=\\\\\\=U\left(U\Big(6\times2\Big)+U\Big(5\Big)+U\Big(1\times9\Big)\right)=\\\\\\=U\left(U\Big(12\Big)+U\Big(5\Big)+U\Big(9\Big)\right)=\\\\=U\left(2+5+9\right)=\\\\=U\left(16\right)=\boxed{\bf6}[/tex]
