If \alpha and \beta are the roots of the equation x^{2}-7x+1=0, then what is the value of \alpha^{4}+\beta^{4}?

Explanation

Sum of roots \alpha+\beta=7 and product \alpha\beta=1. First, \alpha^2+\beta^2 = (\alpha+\beta)^2-2\alpha\beta = 49-2=47. Next, \alpha^4+\beta^4 = (\alpha^2+\beta^2)^2-2(\alpha\beta)^2 = 47^2-2(1)^2 = 2209-2 = 2207.

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