If
(√6 + √2)9 A + B√C
---------- = -------
(√8)9 D
Then, find the integers values of A, B, C, and D with C square-free.
Sometimes I choose a computer solution because it looks like it might be more fun, or faster. But this time, I didn't.
(√6 + √2)^9 = √2^9 * (1 + √3)^9
= 16√2 * (1 + √3)^9
(1 + √3)^9 =
1 + 9√3 + 36*3 + 84*3√3 + 126*9 + 126*9√3 + 84*27 + 36*27√3 + 9*81 + 81√3
= 4240 + 2448√3
(√8)^9 = √8*8^4 = 8192√2
LHS: 16√2 * (4240 + 2448√3)
----------------------
8192√2
LHS: 16^2 * (265 + 153√3)
----------------------
8192
LHS: (265 + 153√3)
-------------
32
A + B√C 265 + 153√3
--------- = ---------
D 32
(A,B,C,D) = (256,153,3,32)
|
|
Posted by Larry
on 2025-03-17 12:44:15 |
Analytic solution
