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