Correct Answer: $22 \sqrt{2}$

Solution : Given: $x^2 -\frac{1}{x^2} = 4\sqrt6$
Squaring both sides we get,
⇒ $x^4 +\frac{1}{x^4} - 2 = 96$
⇒ $x^4 +\frac{1}{x^4}+2 = 98+2$
⇒ $ (x^2+\frac{1}{x^2})^2 =100$
⇒ $x^2 +\frac{1}{x^2} = 10$ ---------------(1)
⇒ $x^2 +\frac{1}{x^2} -2= 10-2$
⇒ $(x-\frac{1}{x})^2=8$
⇒ $(x-\frac{1}{x}) = 2\sqrt2$ ----------------(2)
So, $x^3 -\frac{1}{x^3} = (x-\frac{1}{x})(x^2 +\frac{1}{x^2}+ x×\frac{1}{x})$
⇒ $x^3 -\frac{1}{x^3}= 2\sqrt2 × (10+1)= 22\sqrt2$
Hence, the correct answer is $22\sqrt2$.