If $2 a-left(frac{2}{a} ight)+4=0$, then what is the value of $a^{3}-$ $left(frac{1}{a^{3}} ight)+14 ?$

$2 a-frac{2}{a}+4=0$

$Rightarrow 2left(a-frac{1}{a}+2 ight)=0$

$Rightarrow a-frac{1}{a}=-2$

On cubing both sides,

$left(a-frac{1}{a} ight)^{3}=-8$

$Rightarrow a^{3}-frac{1}{a^{3}}-3left(a-frac{1}{a} ight)=-8$

$Rightarrow a^{3}-frac{1}{a^{3}}-3 imes-2=-8$

$Rightarrow a^{3}-frac{1}{a^{3}}+6+8=0$

$Rightarrow a^{3}-frac{1}{a^{3}}+14=0$