Punctul A
[tex]\displaystyle{ (25^{4})^{3} : (4^{2} + 3^{2})^{11} - (0^{2010} + 1^{2012} + 2013^{0}) = }[/tex]
[tex]\displaystyle{ = 25^{12} : (16 + 9)^{11} - (0 + 1 + 1) }[/tex]
[tex]\displaystyle{ = 25^{12} : 25^{11} - 2 }[/tex]
[tex]\displaystyle{ = 25 - 2 }[/tex]
[tex]\boxed{=23}[/tex]
Punctul B
[tex]\displaystyle{ [123 +2^{7} \cdot 2^{2} +2^{67} :2^{27}]:[123 +2^{14}:2^{5} +(2^{5})^{8}]=}[/tex]
[tex]\displaystyle{=(123 + 2^{9} + 2^{40}):(123+2^{9} + 2^{40}) }[/tex]
[tex]\boxed{=1}[/tex]
Punctul C
[tex]\displaystyle{ (2^{8} + 9^{2}+5^{6} + 36^{2}):(0^{6} +4^{4} + 3^{4} +25^{3} + 6^{4}) = }[/tex]
[tex]\displaystyle{ = [2^{8} + (3^{2})^{2} + 5^{6} + (6^{2})^{2}] : [0 + (2^{2})^{4} + 3^{4} + (5^{2})^{3} + 6^{4}] }[/tex]
[tex]\displaystyle{ =(2^{8} + 3^{4} + 5^{6} + 6^{4}) : (2^{8} + 3^{4} + 5^{6} + 6^{4}) }[/tex]
[tex]\boxed{=1}[/tex]
Punctul D
[tex]\displaystyle{ (4 + 8 + 12 + 16 + ... + 8000) : (2 + 4 + 6 + 8 + ...+4000 ) + ( 2^{53}+2^{52} +2^{51}) : (2^{17})^{3}=}[/tex]4 + 8 + 12 + 16 + ... + 8000 = ?
Ratia = 8 - 4 = 4
Numarul de termeni = (8000 - 4) : 4 + 1 = 7996 : 4 + 1 = 1669 + 1 = 1670
Suma = (8000 + 4) * 1670 : 2 = 8004 : 2 * 1670 = 4002 * 1670
2 + 4 + 6 + 8 + ... + 4000 = ?
Ratia = 4 - 2 = 2
Numarul de termeni = (4000 - 2) : 2 + 1 = 3998 : 2 + 1 = 1999 + 1 = 2000
Suma = (4000 + 2) * 2000 : 2 = 4002 * 1000
(4 + 8 + 12 + 16 + ... + 8000) : (2 + 4 + 6 + 8 + ... + 4000) = 4002 * 1670 : 4002 * 1000 = 1670 * 1000 = 1.670.000
[tex]\displaystyle{ 2^{53} + 2^{52} + 2^{51} = 2^{51} \cdot (2^{2} + 2 + 1) = 2^{51} \cdot 7 }[/tex]
[tex]\displaystyle{ \rightarrow (2^{53} + 2^{52} + 2^{51}):(2^{17})^{3} = 2^{51} \cdot 7 : 2^{51} = 7 }[/tex]
⇒ cele doua paranteze + 7 = 1.670.000 + 7 = 1.670.007
