Răspuns:
Explicație pas cu pas:
a+1/a=2√3 notam a+1/a=t
A=a^4+1/a^4+a³+1/a³+a²+1/a²+a+1/a
(a+1/a)²=a²+2+1/a² ⇒a²+1/a²= (a+1/a)²-2=t²-2
(a+1/a)³=a³+3a·1/a²+3a²·1/a+1/a³ ⇒ a³+1/a³=( a+1/a)³-3( a+1/a)=t³-3t
(a²+1/a²)²=a^4+2+1/a^4 ⇒a^4+1/a^4=(a²+1/a²)²-2=
=(t²-2)²-2=t^4-4²t+4-2=t^4-4t²+2
A=t^4-4t²+2+t³-3t+t²+t-2=t^4+t³-3t²-2t+2=16×9+24√3-36-4√3=
=144-36+20√3=108+20√3=4(27+5√3)
