Question

va rooog ajutatima dau coroana​

Answer 1

Punctul A

$\displaystyle{ (25^{4})^{3} : (4^{2} + 3^{2})^{11} - (0^{2010} + 1^{2012} + 2013^{0}) = }$

$\displaystyle{ = 25^{12} : (16 + 9)^{11} - (0 + 1 + 1) }$

$\displaystyle{ = 25^{12} : 25^{11} - 2 }$

$\displaystyle{ = 25 - 2 }$

$\boxed{=23}$

Punctul B

$\displaystyle{ [123 +2^{7} \cdot 2^{2} +2^{67} :2^{27}]:[123 +2^{14}:2^{5} +(2^{5})^{8}]=}$

$\displaystyle{=(123 + 2^{9} + 2^{40}):(123+2^{9} + 2^{40}) }$

$\boxed{=1}$

Punctul C

$\displaystyle{ (2^{8} + 9^{2}+5^{6} + 36^{2}):(0^{6} +4^{4} + 3^{4} +25^{3} + 6^{4}) = }$

$\displaystyle{ = [2^{8} + (3^{2})^{2} + 5^{6} + (6^{2})^{2}] : [0 + (2^{2})^{4} + 3^{4} + (5^{2})^{3} + 6^{4}] }$

$\displaystyle{ =(2^{8} + 3^{4} + 5^{6} + 6^{4}) : (2^{8} + 3^{4} + 5^{6} + 6^{4}) }$

$\boxed{=1}$

Punctul D

$\displaystyle{ (4 + 8 + 12 + 16 + ... + 8000) : (2 + 4 + 6 + 8 + ...+4000 ) + ( 2^{53}+2^{52} +2^{51}) : (2^{17})^{3}=}$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

$\displaystyle{ 2^{53} + 2^{52} + 2^{51} = 2^{51} \cdot (2^{2} + 2 + 1) = 2^{51} \cdot 7 }$

$\displaystyle{ \rightarrow (2^{53} + 2^{52} + 2^{51}):(2^{17})^{3} = 2^{51} \cdot 7 : 2^{51} = 7 }$

⇒ cele doua paranteze + 7 = 1.670.000 + 7 = 1.670.007