Question : If $x+\left [\frac{1}{(x+7)}\right]=0$, what is the value of $x-\left [\frac{1}{(x+7)}\right]$?

Option 1: $3\sqrt{5}$

Option 2: $3\sqrt{5}-7$

Option 3: $3\sqrt{5}+7$

Option 4: $8$

Question : If $\left(x+\frac{1}{x}\right)=\sqrt{6}$ and $x>1$, what is the value of $\left(x^8-\frac{1}{x^8}\right)$?

Option 1: $120\sqrt{3}$

Option 2: $128\sqrt{3}$

Option 3: $112\sqrt{3}$

Option 4: $108\sqrt{3}$

Question : If $\left(x+\frac{1}{x}\right)=5$, and $x>1$, what is the value of $\left(x^8-\frac{1}{x^8}\right)?$

Option 1: $60605 \sqrt{21}$

Option 2: $60615 \sqrt{21}$

Option 3: $60705 \sqrt{21}$

Option 4: $60725 \sqrt{21}$

Question : If $x-\frac{1}{x}=1$, then what is the value of $\left (\frac{1}{x-1}-\frac{1}{x+1}+\frac{1}{x^{2}+1}-\frac{1}{x^{2}-1} \right)\;$?

Option 1: $\pm \sqrt{5}$

Option 2: $\frac{2}{5}$

Option 3: $\pm\frac{2}{\sqrt{5}}$

Option 4: $\pm\frac{\sqrt{5}}{2}$

Question : If $\left(x^2 - \frac{1}{x^2}\right) = 4 \sqrt{6}$ and $x>1$, what is the value of $\left(x^3 - \frac{1}{x^3}\right)?$

Option 1: $20 \sqrt{2}$

Option 2: $24 \sqrt{2}$

Option 3: $18 \sqrt{2}$

Option 4: $22 \sqrt{2}$