Since the exponent is even, lets get rid of at least one square root:(sqrt(3) + sqrt(2))^1984 = [(sqrt(3) + sqrt(2))^2]^974 = [5 + 2*sqrt(6)]^974
That number's conjugate is [5 - 2*sqrt(6)]^974. The conjugate is positive and less than 1. 5 - 2*sqrt(6) by itself is 0.1010205. Raised to the 974th power leaves a very tiny number with 969 zeroes after the decimal before the first nonzero digit: calculated from log_10([5 - 2*sqrt(6)]^974) = -969.705.
Because the two values are conjugates, the sum [5 + 2*sqrt(6)]^974 + [5 - 2*sqrt(6)]^974 is an integer. Then subtracting [5 - 2*sqrt(6)]^974 will leave [5 + 2*sqrt(6)]^974 showing 969 9's after the decimal point. The 48th digit is well within that range so must be 9.
Solution
