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48th digit in the head (Posted on 2019-01-30) Difficulty: 3 of 5
What is the 48th digit after the decimal point of the number (21/2 + 31/2)1948.

You should solve this problem in your head.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution Solution | Comment 2 of 4 |
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.

  Posted by Brian Smith on 2019-01-31 01:37:48
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