Question

aratati ca namarul n=27^9 × 32^11 ÷ 2 - 16^6 × 2 × 6^27 este patrat perfect

Answer 1

$n=( 3^{3} ) ^{9} *( 2^{5}) ^{11} :2-(2 ^{4} ) ^{6} *2*(2*3) ^{27}$
$n=3^{27} *2^{55} :2-2 ^{24} *2*2 ^{27} *3^{27}$
$n=3^{27} *2^{55-1} -2 ^{24+1+27} *3^{27}$
$n=3^{27} *2^{55-1} -2 ^{24+1+27} *3^{27}$
$n=3^{27} *2^{54} -2 ^{52} *3^{27}$
$n=3^{27} *2^{52} ( 2^{2} -1)$
$n=3^{27} *2^{52} ( 4 -1)$
$n=3^{27} *2^{52} *3$
$n=3^{27+1} *2^{52}$
$n=3^{28} *2^{52}$
$n=3^{28} *2^{52}$
$n=3^{2*14} *2^{2*26}$
Se numeşte pătrat perfect orice număr natural care se poate scrie ca puterea a doua a altui număr natural.

a pătrat perfect $a=k^{2}$

$n=3^{2*14} *2^{2*26}$ ⇒ n este pătrat perfect