sqrt(79 + 24*sqrt(7)) can be written in the form a + b*sqrt(c), where a, b and c are integers and c has no factors which is a perfect square of any positive integer other than 1. Find abc.
+129933 There is an easier way to do this but I've forgotten the method.....here's a clumsier way !!!!
Square both sides
79 + 24sqrt (7) = a^2 + 2absqrt (7) + b^2 * c
This implies that
c = 7
24 = 2ab
12 = ab
Testing some values
a b a^2 b^2 * c a^2 + b^2 * c = 79 ???
1 12 1 144 * 7 No
2 6 4 36 * 7 No
3 4 9 16 * 7 No
4 3 16 9 * 7 Yes
So sqrt [ 79 + 24sqrt 7 ] = sqrt [ ( 4 + 3sqrt 7)^2 ] = 4 + 3sqrt 7
So a = 4 b = 3 and c = 7
abc = 4 * 3 * 7 = 84
