Question

1.Stabiliți dacă urmatoarele numere sunt pătrate perfecte.
9 la puterea 13=
4 la puterea 7=
64 la puterea 25=
25 la puterea 2001=
16 la puterea 35×9 la puterea 25=
4 la puterea 5×49la puterea 11=
3 la puterea 6+3 la puterea 7=
5 la puterea 7+4×5 la puterea 6=
3 la puterea 5+3 la puterea 6-8×3=
2 la puterea 11 + 2 la puterea 12=
3 la puterea 45-3 la puterea 44-3 la puterea 43=
2 la puterea 55-2 la puterea54-2la puterea53=
5 la puterea 21-5 la puterea 20=

2.Arătați ca numărul ,,a" este pătrat perfect
a)a=(1+2+3+.....+99)×2+100=?
b)a=1+2+3+....50-25×50=?
c)a=3+6+9+....+75-75=?
OFER COROANA!

Answer 1

9^13=(3^2)^13=(3^13)^2 e p.p
4^7=(2^2)^7=(2^7)^2 e p.p
64^25=(8^2)^25=(8^25)^2 e p.p
25^2001=(5^2)^2001=(5^2001)^2 e p.p
16^35*9^25=(4^2)^35*(3^2)^25=(4^35)^2*(3^25)^2=(4^35*3^25)^2 e p.p
4^5*49^11=(2^2)^5*(7^2)^11=(2^5)^2*(7^11)^2=(2^5*7^11)^2 e p.p
3^6(1+3)=(3^3)^2*2^2=(3^3*2)^2   e p.p 
5^7+4*5^6=5^6(5+4)=(5^3)^2*3^2=(5^3*3)^2 e p.p
3(3^4+3^5-8)=3(81+243-8)=3*317  nu e p.p
2^11(1+2)=2^11*3 nu e pp
3^43(3^2-3-1)=3^43*5 nu e p.p
2^53(2^2-2-1)=2^53 nu e p.p
5^20(5-1)=(5^10)^2*2^2=(5^10*2)^2 e pp

2. a=99*100/2*2+100=99*100+100=100(99+1)=100^2
b. a=50*51/2-25*50=50(51/2-25)=50(51-50):2=25=5^2 
c. a= 3(1+2+....25)-75=3*25*26/2-75=
75*13-75=75*12=5^2*3*3*4=5^2*3^2*2^2=(5*3*2)^2