Let $\alpha$ and $\beta$ be the roots of the equation $x^2 - 6x - 2 = 0$. If $a_n = \alpha^n - \beta^n$ for $n \ge 1$,then the value of $\frac{a_{10} - 2a_8}{2a_9}$ is equal to:

If the sum of the cubes of the roots of the equation $x^3-ax^2+bx-c=0$ is zero,then $a^3+3c=$ (in $ab$)

If $\alpha, \beta$ are the roots of the equation $x^2 - px + r = 0$ and $\alpha/2, 2\beta$ are the roots of the equation $x^2 - qx + r = 0$,then what is the value of $r$?

If the roots of the equation $ax^2 + bx + c = 0$ are reciprocal to each other,then

If $\alpha$ and $\beta$ are two roots of the equation $x^{2}-64x+256=0$,then the value of $\left(\frac{\alpha^{3}}{\beta^{5}}\right)^{\frac{1}{8}}+\left(\frac{\beta^{3}}{\alpha^{5}}\right)^{\frac{1}{8}}$ is

Let $\alpha, \beta, \gamma$ be the roots of the equation $x^3+px+q=0$ and $f(x)=3p^2x^2+p^2x+3q$. Then $\sum \alpha^2 \beta + \sum \alpha^4 =$