Question

arătați că 2²⁰¹¹-2²⁰¹⁰-2²⁰⁰⁹-2²⁰⁰⁸ este pătrat perfect. ​

Answer 1

Răspuns:

Explicație pas cu pas:

2^2011 - 2^2010 - 2^2009 - 2^2008

= 2^2008*(2^3 - 2^2 - 2 - 1)

= 2^2008*(8 - 4 - 2 - 1)

= 2^2008

= (2^1004)^2 patrat perfect

Answer 2

$2^{2011}-2^{2010}-2^{2009}-2^{2008}=$

$2^{2008}+2^3-2^{2008}+2^2-2^{2008}+2^1-2^{2008}+2^0=$

$2^{2008}\,(2^3-2^2-2-1)=$

$2^{2008} \, (8-4-2-1)=$

$2^{2008}\,(4-2-1)=$

$2^{2008}\,(2-1)=$

$2^{2008}\cdot1=2^{2008}$

$\implies 2^{2008}=2^{1004\cdot2}=(2^{1004})^2$ $- \,patrat\:perfect$