[tex]\displaystyle\\
3a)\\
12^{55}:6^{55}:2^{50} -7^1\cdot 5^0+13^2=\\
=(12^{55}:6^{55}):2^{50} -7^1\cdot 5^0+13^2=\\
=(12:6)^{55}:2^{50} -7^1\cdot 5^0+13^2=\\
=2^{55}:2^{50} -7^1\cdot 5^0+13^2=\\
=2^{55-50} -7^1\cdot 5^0+13^2=\\
=2^{5} -7\cdot 1+13^2=\\
=32 -7+169=\\
=25+169= \boxed{\bf 194}\\\\
3b)\\
(2\cdot 5)^3+45^{45}:9^{45}:5^{44}-2000^0=
=(10)^3+(45:9)^{45}:5^{44}-1=\\
=(10)^3+5^{45}:5^{44}-1=\\
=(10)^3+5^{45-44}-1=\\
=(10)^3+5^{1}-1= 1000+5-1=\boxed{\bf 1004}[/tex]
[tex]\displaystyle\\
3c)\\
15^{38}:5^{38}-\Big(3^{19}\Big)^2=\\
=(15:5)^{38} - 3^{19\times2}=\\
=3^{38} - 3^{38}=\boxed{\bf0}\\\\
3d)\\
=30^{n+3}:10^{n+3}-3^n\cdot 3^3 =\\
=(30:10)^{n+3} - 3^{n+3}=\\
=(3)^{n+3} - 3^{n+3}=\boxed{\bf0}
[/tex]
[tex]\displaystyle\\
4)\\
(4^{50}+4^{49}+4^{48}):21 =\\
=(4^{48+2}+4^{48+1}+4^{48}):21 =\\
=(4^{48}\cdot 4^2+4^{48}\cdot 4^1+4^{48}):21 =\\
=4^{48}\Big(4^2+4^1+1\Big):21 =\\
=4^{48}\Big(16+4+1\Big):21 =\\
=4^{48} \times 21:21 = \boxed{4^{48}}\\
4^{48} = 4^{2\times 24} = \Big(4^{24}\Big)^2 = \texttt{\bf patrat perfect}\\
4^{48} = 4^{3\times 16} = \Big(4^{16}\Big)^3 = \texttt{\bf cub perfect}\\[/tex]
[tex]\displaystyle\\
5)\\
U(S) = U\Big[ U(8^{283})+U(9^{126})\Big]=\\\\
= U\Big[ U(8^{280+3})+U(9^{2\times 63})\Big]=\\\\
= U\Big[ U(8^{280} \times8^3})+U(9^{2\times 63})\Big]=\\\\
= U\Big[ U(8^{4\times 70} \times8^3})+U(9^{2\times 63})\Big]=\\\\
= U\Big[ U\Big(\Big(8^4\Big)^{70} \times8^3}\Big)+U\Big(\Big(9^2\Big)^{63}\Big)\Big]=\\\\
= U\Big[ U\Big(4096^{70} \times 512}\Big) +U\Big(81^{63}\Big)\Big]=\\\\
= U\Big[ U\Big(6 \times 2}\Big) +U\Big(1^{63}\Big)\Big]=\\\\
= U[ U(12+1)]=\boxed{\bf 3}[/tex]
[tex]\displaystyle\\
6)\\
~[(x+360:4)\cdot 5+700]:600 = 2\\
~[(x+90)\cdot 5+700] = 2\times 600\\
(x+90)\cdot 5+700 = 1200\\
(x+90)\cdot 5 = 1200-700\\
(x+90)\cdot 5 = 500\\
x+90 = 500:5\\
x+90 = 100\\
x = 100 - 90 = \boxed{\bf10}
[/tex]
