Rules: 
Normal sudoku rules apply. 
The white dot connects two consecutive numbers. 
Each blue line can be read from one end to the other as an equation of the form A^B=C. 
If A, B or C has multiple digits, they are read along the line in the same direction as the equation.
Examples (for a 4-digit line): 
8264 - valid, since 8^2=64 
4628 - valid (same equation from other end) 
8246 - not valid (64 is backwards) 
6482 - not valid (64=8^2, but order is not A^B=C) 
This is the first and harder of a pair of puzzles based on the powerlines rule. The other (GAS Powerlines) is much easier, but I think this one is nicer, and there should be more satisfaction/challenge in tackling it first (but for anyone who gets stuck, I recommend trying that one). 
Solve online:
CTC
F-Puzzles