Question

Aratati ca urmatoarele numere sunt patrate perfecte:
a) 32 * 93 - 32 * 85 =
b) 2^83 - 4^41 =
c) 11 * ( 5^21-5^20-5^18) =

Answer 1

$a) 32\cdot93-32\cdot85 = 32(93-85)=32\cdot8 = 8\cdot4\cdot8 = 8^2\cdot2^2=16^2=256 \\ =\ \textgreater \ 256 p.p.$

$b)2^{83}-4^{41}= 2^{83}-(2^2)^{41}= 2^{83}-2^{82}= 2^{82}(2-1)= 2^{82}=(2^{41})^{2} \\ =\ \textgreater \ (2^{41})^{2} p.p.$

$c) 11\cdot(5^{21}-5^{20}-5^{18})= 11\cdot5^{18}\cdot(5^3-5^2-1)=$$11\cdot5^{18}\cdot(125-25-1)= 11\cdot5^{18}\cdot99= 11\cdot5^{18}\cdot11\cdot9= 11^2\cdot(5^9)^2\cdot3^2$$=(11\cdot5^9\cdot3)^2 \\ =\ \textgreater \ (11\cdot5^9\cdot3)^2 p.p.$