Final answer:
The expression 81a⁴−256 can be factored as a difference of squares, resulting in the factored form (9a² + 16)(9a² − 16), making option A correct.
Step-by-step explanation:
To factor the expression 81a⁴−256 completely, we recognize that both 81 and 256 are perfect squares, and a⁴ is also a perfect square (being (a²)²). The expression can be rewritten as (9a²)² − 16², which is a difference of squares. A difference of squares can be factored as (A+B)(A−B), where A and B are the square roots of the first and second terms respectively.
Hence, (9a²)² − 16² = (9a² + 16)(9a² − 16), which is the product of a binomial and its conjugate. Thus, option A) (9a² + 16)(9a² − 16) is the correct option in the final answer.
To factor the expression 81a⁴−256 completely, we can use the difference of squares formula, which states that a² - b² can be factored as (a + b)(a - b). In this case, we have 81a⁴ = (9a²)² and 256 = 16².
So, the expression can be written as (9a² + 16)(9a² - 16). Therefore, the correct option is A) (9a² + 16)(9a² - 16).