Final answer:
To factor the expression 9x⁴ - 225y⁸, it's recognized as a difference of squares. The expression factors into (3)(x² - 5y⁴)(3x² + 15y⁴), after identifying each term as a square and applying the difference of squares formula.
Step-by-step explanation:
The student is asked to completely factor the expression 9x⁴ - 225y⁸. This is a difference of squares problem, where each term is a perfect square itself. The general formula for the difference of squares is a² - b² = (a - b)(a + b). In this case, the first term 9x⁴ can be written as (3x²)² and the second term 225y⁸ can be written as (15y⁴)². Using the difference of squares formula, the expression can be factored as:
(3x² - 15y⁴)(3x² + 15y⁴)
To provide further simplification, we can factor out a 3 from both terms in the first parenthesis, which gives us:
(3)(x² - 5y⁴)(3x² + 15y⁴)
However, since x² - 5y⁴ does not factor further, our final factored expression is:
(3)(x² - 5y⁴)(3x² + 15y⁴)
This demonstrates the process of factoring by difference of squares and also shows that math provides many paths to simplify and understand expressions.