Multiple choice

If $\alpha$, $\beta$ are the roots of the equation $ { x }^{ 2 }+x+2=0$, then what is $\dfrac { { \alpha }^{ 10 }+{ \beta }^{ 10 } }{ { \alpha }^{ -10 }+{ \beta }^{ -10 } } $ equal to?

  1. $4096$
  2. $2048$
  3. $1024$
  4. $512$
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C Correct answer
Explanation

Given x^2 + x + 2 = 0, roots are alpha, beta. alpha + beta = -1, alpha * beta = 2. Expression = (alpha^10 + beta^10) / ((alpha^10 + beta^10) / (alpha^10 * beta^10)) = (alpha * beta)^10 = 2^10 = 1024.

AI explanation

From the given equation, the sum of the roots alpha plus beta equals -1 and the product alpha beta equals 2. We can simplify the base expression by multiplying the numerator and denominator by alpha to the power of 10 times beta to the power of 10, which gives the new fraction as alpha to the power of 10 times beta to the power of 10. Since alpha times beta equals 2, this expression equals 2 to the power of 10. Calculating 2 to the power of 10 gives 1024.

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