HISTORY OF MATH HOMEWORK 5 SOLUTIONS 9.30, 32, 35, 37 ...

HISTORY OF MATH HOMEWORK 5 SOLUTIONS 3
9.38 Express [52 + √ −2209] 1/3 in the form a + b √ −1.
Note that 2209 = 47 2 . So we must find a, b such that
Working out the cube gives two equations
(a + b √ −1) 3 = 52 + 47 √ −1.
a 3 − 3ab 2 = 52, 3a 2 b − b 3 = 47.
Adding these gives
(a − b)(a 2 + 4ab + b 2 ) = 99.
Subtracting gives
(a + b)(a 2 − 4ab + b 2 ) = 5.
Now let’s be optimistic and look for integer solutions for a, b. The latter equation says that a + b
divides 5, so a + b = ±1, ±5. We find that a + b = 5 leads to
a 2 − 4a(5 − a) + (5 − a) 2 = 1,
which has solutions a = 1, 4, hence b = 4, 1. But the first equation says that a − b > 0, so we
take a = 4, b = 1, and note that this satisfies both equations. Hence
[52 + √ −2209] 1/3 = 4 + √ −1.
Another way to do this, without guessing, is to reconstruct the cubic for which [52+ √ −2209] 1/3
arises in the Cardano formula. We have q = 52 and p 3 = −2209 − q 2 = −17 3 so p = −17. The
cubic
x 3 − 51x = 104
has the root r = 8. So
8 = [52 + √ −2209] 1/3 + [52 − √ −2209] 1/3 .
The quadratic
Y 2 − rY − p = Y 2 − 8Y + 17
has roots 4 ± √ −1. Cubing these, we find that (4 + √ −1) 3 = 52 + √ −2209.