Question

aratati ca B=$2^{6n+7} - 2^{6n+5} + 2^{6n+2}$
se poate scrie ca suma de 4 cuburi perfecte

Answer 1

        
$2^{6n+7} - 2^{6n+5} + 2^{6n+2} = \\ = 2^{6n}* 2^{7} - 2^{6n}* 2^{5} + 2^{6n}* 2^{2}= \\ =2^{6n}(2^{7}-2^{5}+ 2^{2} )= \\=2^{6n}(128-32+ 4)= \\ =2^{6n}*(100)= \;\;\;\text{Descompunem nr. 100 in suma de 4 cuburi perfecte}\\=2^{6n}*(1 + 8 + 27 + 64)= \\ 2^{6n} + 8*2^{6n}+27*2^{6n}+64*2^{6n}= \\ = \boxed{(2^{2n})^{3}+(2*2^{2n})^{3}+(3*2^{2n})^{3}+(4*2^{2n})^{3}}$