In base $10,$ $44 \times 55$ does not equal $3506.$ In what base does $44 \times 55 = 3506$?
Let b be the unkown base
We have that
(4b + 4) ( 5b + 5) = 3b^3 + 5b^2 + 0b + 6 simplify
20b^2 + 40b + 20 = 3b^3 + 5b^2 + 6
3b^3 - 15b^2 - 40b - 14 = 0
Solving this using the Rational Zeroes Theorem shows that the integer solution for b = 7
Proof
447 * 557 =
(4 * 7 + 4) ( 5 * 7 + 5) =
32 * 40 = 128010
And
35067 = 3*(7)^3 + 5*(7)^2 + 0*7 + 6 = 128010
Also converting 1280 from base 10 to base 7
1280 = 182 * 7 + R 6
182 = 26 * 7 + R 0
26 = 3 * 7 + R 5
3 = 0 * 7 + 3
Reading the remainders from bottom to top we have 3506