Final answer:
The expression 625x8 - 1 is factored completely as (25x4 + 1)(5x2 + 1)(5x2 - 1) by recognizing it as a difference of squares and applying the factorization method recursively.
Step-by-step explanation:
To factor the expression 625x8 - 1 completely, we recognize it as a difference of squares. A difference of squares can be factored as (a2 - b2) = (a + b)(a - b). In this case, 625x8 is a perfect square, being (25x4)2, and 1 is also a perfect square, being (1)2. Therefore, the expression can be written as (25x4 + 1)(25x4 - 1). The second factor is again a difference of two squares and can be further factored into (5x2 + 1)(5x2 - 1). Finally, we have factored 625x8 - 1 completely into (25x4 + 1)(5x2 + 1)(5x2 - 1).