Question

scrieti ca diferenta de doua patrate perfecte : a) 21^21 ; b) 53^53 ; c) 9^99 ; d) 55^55 ; e)65^65 ; f)32^31

Answer 1

$a)~21^{21}=21*21^{20}=$

$=(25-4)*(21^{10})^{2}=$

$=5^{2}*(21^{10})^{2}-2^{2}*(21^{10})^{2}$

$=(5*21^{10})^{2}-(2*21^{10})^{2}$


$b)~53^{53}=53*53^{52}=$

$=(64-9)*(53^{26})^{2}=$

$=8^{2}*(53^{26})^{2}-3^{2}*(53^{26})^{2}=$

$=(8*53^{26})^{2}-(3*53^{26})^{2}$


$c)~9^{99}=9*9^{98}=$

$=(25-16)*(9^{49})^{2}=$

$=5^{2}*(9^{49})^{2}-4^{2}*(9^{49})^{2}=$

$=(5*9^{49})^{2}-(4*9^{49})^{2}$


$d)~55^{55}=55*55^{54}=$

$=(64-9)*(55^{27})^{2}=$

$=8^{2}*(55^{27})^{2}-3^{2}*(55^{27})^{2}=$

$=(8*55^{27})^{2}-(3*55^{27})^{2}$


$e)~65^{65}=65*65^{64}=$

$=(81-16)*(65^{32})^{2}=$

$=9^{2}*(65^{32})^{2}-4^{2}*(65^{32})^{2}=$

$=(9*65^{32})^{2}-(4*65^{32})^{2}$


$f)~32^{31}=32*32^{30}=$

$=(36-4)*(32^{15})^{2}=$

$=6^{2}*(32^{15})^{2}-2^{2}*(32^{15})^{2}=$

$=(6*32^{15})^{2}-(2*32^{15})^{2}$