Which of the following correctly factors the expression `25x^4 -16 y^8`?

Options

A. `(25-16)(x^2-y^4)(x^4+y^4)`
B. `(5x^2-4y^4)(5x^2+4y^4)`
C. `(25x^2-y^4)(x^2+16 y^4)`
D. `(5x^4-4y^8)(5x^4+4y^8)`

Answer

To factor the expression \( 25x^4 - 16y^8 \), we can follow these steps:

### Step 1: Identify the structure of the expression
The expression \( 25x^4 - 16y^8 \) is a difference of squares, which can be represented as \( a^2 - b^2 \).

### Step 2: Rewrite the terms as squares
We can rewrite \( 25x^4 \) and \( 16y^8 \) as:
- \( 25x^4 = (5x^2)^2 \)
- \( 16y^8 = (4y^4)^2 \)

So, we have:
\[ 25x^4 - 16y^8 = (5x^2)^2 - (4y^4)^2 \]

### Step 3: Apply the difference of squares formula
The difference of squares can be factored using the formula \( a^2 - b^2 = (a - b)(a + b) \). Here, let \( a = 5x^2 \) and \( b = 4y^4 \).

Thus, we can factor the expression as:
\[ (5x^2 - 4y^4)(5x^2 + 4y^4) \]

### Conclusion
The factored form of the expression \( 25x^4 - 16y^8 \) is:
\[ (5x^2 - 4y^4)(5x^2 + 4y^4) \]