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Let alpha and beta be the roots of x^2 -...

Let `alpha and beta` be the roots of `x^2 - 5x - 1 = 0` then the value of `(alpha^15 + alpha^11 + beta^15 + beta^11)/(alpha^13 + beta^13)` is

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`(alpha ^(15)+ alpha^(11) + alpha^(15) + beta^(11))/(alpha^(13) + beta^(13))`.
= `(alpha ^(15)+ alpha^(15) + alpha^(2) beta^(2)(alpha^(11)+beta^(2)))/(alpha^(13) + beta^(13))" "( because alpha beta = -1)`
`(alpha ^(15)+ alpha^(13) + beta^(2)beta^(15)+alpha^(2)beta^(13))/(alpha^(13) + beta^(13))`
`((alpha ^(13)+ alpha^(13)) ( alpha^(2)+beta^(2)))/(alpha^(13) + beta^(13))`
`alpha^(2) + beta^(2) = (alpha + beta)^(2) - 2alpha beta = 27`
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