Correct Answer: $992 \sqrt{15}$

Solution : $(x-\frac{1}{x})=\sqrt{6}$
Squaring both sides,
⇒ $(x^{2} +\frac{1}{x^2}-2)=6$
⇒ $(x^{2} +\frac{1}{x^2})=8$
Squaring both sides,
⇒ $(x^{4}+\frac{1}{x^4}+2)=64$
⇒ $(x^{4}+\frac{1}{x^4}) = 62$
Squaring both sides,
⇒ $(x^{8}+\frac{1}{x^8}+2)=3844$
⇒ $(x^{8}+\frac{1}{x^8})=3842$
⇒ $(x^{8}+\frac{1}{x^8})^2=3842^2$
⇒ $(x^{8}-\frac{1}{x^8})^2=(x^{8}+\frac{1}{x^8})^2 - 4$
⇒ $(x^{8}-\frac{1}{x^8})^2=3842^2 - 2^2$
⇒ $(x^{8}-\frac{1}{x^8})^2=(3842 +2)(3842-2)$
⇒ $(x^{8}-\frac{1}{x^8})^2=3844 \times 3840$
⇒ $(x^{8}-\frac{1}{x^8})^2=4 × 31 × 31 × 4 × 4 × 4 × 4 × 15$
⇒ $(x^{8}-\frac{1}{x^8}) = 31 × 2 × 4 × 4 × \sqrt{15}$
Hence, the correct answer is $992 \sqrt{15}$.