$\underline{\mathbf{Answer:3}}$

$\underline{\mathbf{Solution:}}$

Let $\alpha$ and $\beta$ are the roots of the equation $\mathrm{ax^2 + bx + c = 0}$

then

$\mathrm {B = \alpha^n}\;\text{[Given]}$

Also we know that:

$\text{Sum of roots} = -\dfrac{b}{a}$

$\text{Product of roots} = \mathrm{\dfrac{c}{a}}$

$\therefore \alpha = \mathrm{\frac{c}{a}}\Rightarrow \alpha\times \alpha^n =\alpha^{n+1}= \dfrac{c}{a} \Rightarrow \alpha = (\frac{c}{a})^{\frac{1}{n+1}}$

then,

$\alpha + \beta = -\frac{b}{a} \Rightarrow \alpha + \alpha^n= -\frac{b}{a}$

Now we can write the quadratic equation $\mathrm{ax^2+bx+c = 0}$ using one of the root $\alpha$ as:

$\mathrm{a\alpha^2 + b\alpha + c = 0}$

$\Rightarrow \mathrm{a\alpha +b + \frac{c}{\alpha}}$

$\mathrm{\Rightarrow \frac{c}{\alpha} + a\alpha + b = 0}$

$\mathrm{\dfrac{c}{(\frac{c}{a})^{\frac{1}{n+1}}}+a\left (\frac{c}{a} \right )^{\frac{1}{n+1}}+b = 0}$

$\mathrm{c \left (\frac{a}{c} \right )^{\frac{1}{n+1}}+a\left ( \frac{c}{a}\right)^{\frac{1}{n+1}}+b = 0}$

$\mathrm{\left ( \frac{c^{n+1}a}{c}\right)^{\frac{1}{n+1}}+ \left ( \frac{a^{n+1}c}{a}\right)^{\frac{1}{n+1}} + b = 0}$

$\mathrm{\Rightarrow (ac^n)^{\frac{1}{n+1}} + (a^nc)^{\frac{1}{n+1}} + b = 0}$

$\therefore \;\mathrm{Option}\; 3$ is the correct answer.