Question : If $\left(x+\frac{1}{x}\right)=5 \sqrt{2}$, and $x>1$, what is the value of $\left(x^6-\frac{1}{x^6}\right) ?$

Option 1: $22970 \sqrt{23}$

Option 2: $23030 \sqrt{23}$

Option 3: $23060 \sqrt{23}$

Option 4: $22960 \sqrt{23}$

Question : If $x^2+y^2=427$ and $xy=202$, then find the value of $\frac{x+y}{x-y}$.

Option 1: $\sqrt{\frac{835}{23}}$

Option 2: $\sqrt{\frac{830}{29}}$

Option 3: $\sqrt{\frac{831}{23}}$

Option 4: $\sqrt{\frac{830}{23}}$

Question : If $x^2-5\sqrt{5}x+1=0$ and $x > 0$, what is the value of $(x^3-\frac{1}{x^3})$?

Option 1: 1331

Option 2: 1364

Option 3: 1296

Option 4: 1244

Question : If $x^3=270+y^3$ and $x=(6+y)$, then what is the value of $(x+y)? $(given that $x>0$ and $y>0$)

Option 1: $2 \sqrt{3}$

Option 2: $\sqrt{3}$

Option 3: $4 \sqrt{3}$

Option 4: $4 \sqrt{2}$

Question : If $x=(\sqrt{6}-1)^{\frac{1}{3}}$, then the value of $\left(x-\frac{1}{x}\right)^3+3\left(x-\frac{1}{x}\right)$ is:

Option 1: $\frac{2 \sqrt{6}-6}{5}$

Option 2: $\frac{4 \sqrt{6}-6}{5}$

Option 3: $\frac{4 \sqrt{6}-6}{3}$

Option 4: $\frac{4 \sqrt{3}-6}{5}$