Discussion
In a quadratic function, the value of the product of the roots (α, β) is 4. Find the value of
\[ \frac{\alpha^{n} + \beta^{n}}{\alpha^{-n} + \beta^{-n}} \]
Answer: B
The value is \(4^{n}\).
Rewrite the denominator over a common denominator:
\[ \begin{aligned} \alpha^{-n} + \beta^{-n} &= \frac{\alpha^{n} + \beta^{n}}{(\alpha\beta)^{n}} \\ \frac{\alpha^{n} + \beta^{n}}{\alpha^{-n} + \beta^{-n}} &= (\alpha\beta)^{n} = 4^{n} \end{aligned} \]
Only the product of the roots matters, which is why the question gives nothing else. \(n^{4}\) swaps base and exponent. \(2^{2n-1} = 4^{n}/2\) and \(4^{n-1} = 4^{n}/4\), so both fall short of the correct value by a constant factor.
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