If one root of the quadratic equation ax^2 + | vedclass

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(B) Let the roots of the quadratic equation $ax^2 + bx + c = 0$ be $\alpha$ and $\alpha^n$. From the relation between roots and coefficients,we have:

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$-b$

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(B) Let the roots of the quadratic equation $ax^2 + bx + c = 0$ be $\alpha$ and $\alpha^n$. From the relation between roots and coefficients,we have: $\alpha + \alpha^n = -\frac{b}{a}$ and $\alpha \cdot \alpha^n = \alpha^{n+1} = \frac{c}{a}$. We need to evaluate the expression $E = (ac^n)^{1/(n+1)} + (a^nc)^{1/(n+1)}$. Substituting $c = a\alpha^{n+1}$: $E = (a(a\alpha^{n+1})^n)^{1/(n+1)} + (a^n(a\alpha^{n+1}))^{1/(n+1)}$ $E = (a^{n+1} \alpha^{n(n+1)})^{1/(n+1)} + (a^{n+1} \alpha^{n+1})^{1/(n+1)}$ $E = a\alpha^n + a\alpha$ $E = a(\alpha^n + \alpha)$ Substituting $\alpha^n + \alpha = -\frac{b}{a}$: $E = a(-\frac{b}{a}) = -b$.

If one root of the quadratic equation $ax^2 + bx + c = 0$ is equal to the $n$th power of the other,then $(ac^n)^{1/(n+1)} + (a^nc)^{1/(n+1)} =$

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