Multiple choice

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

  1. $b$
  2. $-b$
  3. $\dfrac{1}{b^{n+1}}$
  4. $-b^{n+1}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If roots are r and r^n, then r * r^n = c/a and r + r^n = -b/a. The expression a^(n/(n+1)) * c^(1/(n+1)) + c^(n/(n+1)) * a^(1/(n+1)) simplifies to (a*c)^(1/(n+1)) * (a^(n/(n+1)) + c^(n/(n+1))). This is a known identity related to the roots of ax^2+bx+c=0.

AI explanation

Let the roots be r and r^n, so their sum is -b/a and their product is c/a, giving r^(n+1) = c/a. The expression a^(n/(n+1)) * c^(1/(n+1)) simplifies to a * (c/a)^(1/(n+1)), which equals a * r. Similarly, c^(n/(n+1)) * a^(1/(n+1)) evaluates to a * r^n, making the overall expression equal to ar + ar^n. Factoring out a gives a(r + r^n), which is precisely -b because r + r^n equals -b/a.

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