Which of the following expressions is equivalent to `25 x ^(2) y ^(4)-1` ?

Options

A. `5 (x ^(2) y ^(4) -1)`
B. `-5 (xy ^(2) +1 )`
C. `(5xy-1) (5xy+1)`
D. `(5xy^(2) -1) (5xy^(2) +1)`

Explanation

To solve the expression \( 25x^2y^4 - 1 \) and find its equivalent form, we can use the difference of squares formula, which states that \( a^2 - b^2 = (a + b)(a - b) \).

### Step-by-Step Solution:

1. **Identify the terms**:
We can rewrite \( 25x^2y^4 - 1 \) in the form of \( a^2 - b^2 \).
Here, \( a^2 = 25x^2y^4 \) and \( b^2 = 1 \).

2. **Find \( a \) and \( b \)**:
- To find \( a \), we take the square root of \( 25x^2y^4 \):
\[ a = \sqrt{25x^2y^4} = 5xy^2 \]
- To find \( b \), we take the square root of \( 1 \):
\[ b = \sqrt{1} = 1 \]

3. **Apply the difference of squares formula**:
Now we can apply the formula \( a^2 - b^2 = (a + b)(a - b) \):
\[ 25x^2y^4 - 1 = (5xy^2 + 1)(5xy^2 - 1) \]

4. **Final Expression**:
Thus, the expression \( 25x^2y^4 - 1 \) can be factored as:
\[ (5xy^2 + 1)(5xy^2 - 1) \]

### Answer:
The equivalent expression for \( 25x^2y^4 - 1 \) is \( (5xy^2 + 1)(5xy^2 - 1) \).