Surds & Indices

If (a + 1/a) = 7√3, then what is the value of (a^6 + a^(-6))?...

If (x + 1/x) = 7, x > 0. Find the positive value of (x - 1/x)....

If $x + \frac{1}{x} = 2$, then find the value of $$ x^{2723} + \frac{1}{x^{3356}}. $$...

If x = 7 - 4√3, then find the value of (x² + 1/x²)....

If x > 1 and x² + 1/x² = 83, then find the value of x³ - 1/x³....