If `alphaandbeta` are the roots of the equation `x^(2)-6x+12=0` and the value of `(alpha-2)^(24)-((beta-6)^(8))/(alpha^(8))+1` is `4^(a)`, then the value of a is ______.

Answer

12

Explanation

We have `alpha^(2)-6alpha+12=0`
`:.6(alpha-2)=alpha^(2)` and
`beta^(2)-6beta+12=0`
`:.beta(beta-6)+12=0`
So, `(alpha-2)^(24)-((beta-6)^(8))/(alpha^(8))+1=(alpha-2)^(24)-(12^(8))/(alpha^(8)beta^(8))+1`
`=(alpha-2)^(24)-(12^(8))/(12^(8))+1`
`=(alpha-2)^(24)`
`=(3pmsqrt3i)^(24)(alpha=3pmsqrt3i)`
`=(1pmsqrt3i)^(24)`
`=(2(cos""(pi)/(3)pmisin""(pi)/(3)))^(24)`
`=2^(24)(cos""8pi+isin8pi)=4^(12)`