Problem:
The real root of the equation 8x3−3x2−3x−1=0 can be written in the form c3a+3b+1, where a,b, and c are positive integers. Find a+b+c.
Solution:
The equation 8x3−3x2−3x−1=0 is equivalent to x3+3x2+3x+1=9x3. Thus (x+1)3=9x3, so x=39−11. Multiplying numerator and denominator by (39)2+39+1 yields x=8381+39+1. The requested sum is 81+9+8=98.