Question

Aratati ca urmatoarele numere sunt patrate perfecte :
Ex:A) 64·56 64·(56 8)=64·64=64²
b)32·93-32·85=
c)2⁸³-4⁴¹=
d)11·(5²¹-5²⁰-5¹⁸)=
Ma puteti ajuta ? Va rog !!

Answer 1

     
$a) \\ Este\;exemplu \;de \;rezolvare. \\ \\ b)\;\;\; \\ 32\cdot 93 -32\cdot 85=32(93-85) = \\ =32 \cdot 8 = 2^5 \cdot 2^3=2^{5+3}=2^8=\boxed{(2^4)^2}=\boxed{16^2} \\ \\ c) \\ 2^{83}-4^{41}= 2^{83}-(2^2)^{41}=2^{83}-2^{2 \; \cdot \;41}= 2^{83}-2^{82}= \\ =2^{82}(2^1-2^0)=2^{82}(2-1)=2^{82} \cdot 1=2^{82}=2^{41\;\cdot \;2}=\boxed{(2^{41})^2}$

$d) \\ 11(5^{21}-5^{20}-5^{18})=11 \cdot5^{18}(5^3-5^2-1)=11 \cdot5^{18}(125-25-1)= \\ =11 \cdot5^{18}\cdot 99=11 \cdot5^{18}\cdot 11 \cdot 9=11^2 \cdot 5^{18} \cdot 3^2=\boxed{(3 \cdot 5^9 \cdot 11)^2}$