If one root of the quadratic equation $a x^2+b x+c=0$ is equal to the $n$th power of the other, then $\left(a c^n\right)^{1 / n+1}+\left(a^n c\right)^{1 / n+1}=$

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Aicharya · Accepted Answer

Let $\alpha$ and $\alpha^n$ are the roots of the equation $ \begin{aligned} \alpha+ & \alpha^n=-\frac{b}{a} 	ext { and } \alpha^{n+1}=\frac{c}{a} \ & =\left(a c^night)^{\frac{1}{n+1}}+\left(a^n cight)^{\frac{1}{n+1}} \ & =\left[a\left(a \alpha^{n+1}ight)^night]^{\frac{1}{n+1}}+\left(a^n 	imes a \alpha^{n+1}ight)^{\frac{1}{n+1}} \ & =\left[a^{n+1} \alpha^{n(n+1)}ight]^{\frac{1}{n+1}}+\left(a^{n+1} \alpha^{n+1}ight)^{\frac{1}{n+1}} \ & =a \alpha^n+a \alpha=a\left(\alpha^n+\alphaight) \ & a\left(-\frac{b}{a}ight)=-b \end{aligned} $

If one root of the quadratic equation $a x^2+b x+c=0$ is equal to the $n$th power of the other, then…

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