Question

Aratati ca numarul x=[2^30^2 *(2^6)^100*2+(32^4)^100 :2^500]^2+2^3004 este patrat perfect

Answer 1

x = [2^30^2 ×(2^6)^100 × 2 + (32^4)^100 : 2^500]^2 + 2^3004 =

= {2^900 × 2^600 × 2 + [(2^5)^4]^100 : 2^500}^2 + 2^3004 =

= [2^1501 + (2^20)^100 : 2^500]^2 + 2^3004 =

= (2^1501 + 2^2000 : 2^500)^2 + 2^3004 =

= (2^1501 + 2^1500)^2 + 2^3004 =

= [2^1500×(2+1)]^2 + 2^3004 =

= 2^3000 × 9 + 2^3004 =

= 2^3000 × (9 + 2^4) =

= 2^3000 × 25 =

= (2^1500)^2 × 5^2 =

= (2^1500 × 5)^2, deci numărul x este pătrat perfect