Question

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

• A. b
• B. -b
• C.
• D.

Answer 1

The correct option is B

-b

Let $\alpha$, ${\alpha }^n$ be two roots,

Then $\alpha +{\alpha }^n$ = -$\frac{b}{a}$, $\alpha {\alpha }^n$ = $\frac{c}{a}$

Eliminating $\alpha$, we get ($\frac{c}{a}{)}^{\frac{1}{n+1}}$ + ($\frac{c}{a}{)}^{\frac{n}{n+1}}$ = -$\frac{b}{a}$

⇒ $a.a^{-\frac{1}{n+1}}$. $c^{\frac{1}{n+1}}$ + $a.a^{-\frac{n}{n+1}}$. $c^{\frac{n}{n+1}}$ = -b

or $(a^{n}c{)}^{\frac{1}{n+1}}$ + $(a{c}^n{)}^{\frac{1}{n+1}}$ = -b