Simplify: `[2x^(2) - (1)/(400)y^(2)]^(2)- [2x^(2)+ (1)/(400) y^(2)]^(2)`

Options

A. `-(x^(2)y^(2))/(40)`
B. `-(x^(2)y^(2))/(50)`
C. `(xy)/(50)`
D. `-(x^(2)y^(2))/(5)`

Answer

To simplify the expression \([2x^{2} - \frac{1}{400}y^{2}]^{2} - [2x^{2} + \frac{1}{400}y^{2}]^{2}\), we can follow these steps:

### Step 1: Identify the terms
Let:
- \( a = 2x^{2} \)
- \( b = \frac{1}{400}y^{2} \)

Now, we can rewrite the expression as:
\[ (a - b)^{2} - (a + b)^{2} \]

### Step 2: Apply the difference of squares formula
We can use the identity:
\[ A^{2} - B^{2} = (A - B)(A + B) \]
where \( A = (a - b) \) and \( B = (a + b) \).

Thus, we have:
\[ [(a - b) - (a + b)][(a - b) + (a + b)] \]

### Step 3: Simplify the first part
Calculating \( (a - b) - (a + b) \):
\[ (a - b) - (a + b) = a - b - a - b = -2b \]

### Step 4: Simplify the second part
Calculating \( (a - b) + (a + b) \):
\[ (a - b) + (a + b) = a - b + a + b = 2a \]

### Step 5: Combine the results
Now we combine the results from Step 3 and Step 4:
\[ (-2b)(2a) = -4ab \]

### Step 6: Substitute back the values of \( a \) and \( b \)
Substituting back \( a = 2x^{2} \) and \( b = \frac{1}{400}y^{2} \):
\[ -4(2x^{2})\left(\frac{1}{400}y^{2}\right) \]

### Step 7: Simplify the expression
Calculating:
\[ -4 \cdot 2 \cdot \frac{1}{400} \cdot x^{2}y^{2} = -\frac{8}{400}x^{2}y^{2} = -\frac{1}{50}x^{2}y^{2} \]

### Final Answer
Thus, the simplified expression is:
\[ -\frac{1}{50}x^{2}y^{2} \]

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