4 to the power of 521 or 5 to the power of 421

Did you try it on a calulator?

4^521=4.7125 X 10^313, and

5^421=1.8466 X 10^294

I leave it for you to decide which is the bigger number!

Here's another way to see this :

Let's assume that they are equal and show by contradiction that this can't be true.........

Let 5^421  = x     and    4^521 = x    take the log of both sides of these and we have :

log(5)^421  = x                                    log(4)^521 = x

And we can write

421log(5)  = log (x)                                521log(4) = log (x)

294.26  = log(x)                                     313.67 = log (x)

And it's clear that the expression on the right is larger than the expression on the left....i.e., 4^521 > 5^421