1. [tex] 5^{34} +5^{17}= 5^{17}(5^{2} +1)= 5^{17}*26; \sqrt{5^{17}*26} = 5^{4} \sqrt{5*26}= 5^{4} *11,4 [/tex] nu este patrat perfect
[tex]2. \frac{25}{47} + \frac{49}{17}= \frac{25*17+49*47}{47*17} = \frac{425+2303}{799} = \frac{978775}{799}= 1225 \\ \\ \sqrt{1225}= 35 \\ \\ 1225=35^{2} [/tex]
deci [tex] \frac{25}{47} + \frac{49}{17} [/tex] [tex] =1225=35^{2} [/tex]- patrat perfect
3. 2+4+6+....+100=2(1+2+3+4+....+50)=[50×51]:2=1275=35,707 nu este patrat perfect
(1+2+3+4+....+50)-suma Gauss cu formula de calcul S=[n(n+1)]:2
4.Arătați ca nr 1+3+5+.....+99 este P.P.
1+3+5+.....+99 -progresie aritmetica, an=99, a1=1, ratia=2
an=a1+(n-1)×r
99=1+(n-1)×2
99-1+2=2n
100=2n
n=50
S=[(a1+an)×n]:2=[(1+99)×50]:2=(100×50):2=50×50=50²
deci 1+3+5+.....+99=50×50=50² - patrat perfect
