IF `(10^12+25)^2-(10^12-25)^2=10^n` then the value of n is

Options

A. 20
B. 14
C. 10
D. 5

Answer

To solve the equation \((10^{12} + 25)^2 - (10^{12} - 25)^2 = 10^n\), we can use the difference of squares formula.

### Step-by-step Solution:

1. **Identify the Expression**:
We have the expression \((10^{12} + 25)^2 - (10^{12} - 25)^2\).

2. **Apply the Difference of Squares Formula**:
The difference of squares states that \(a^2 - b^2 = (a - b)(a + b)\). Here, let:
- \(a = 10^{12} + 25\)
- \(b = 10^{12} - 25\)

So, we can rewrite the expression as:
\[ (10^{12} + 25 - (10^{12} - 25))((10^{12} + 25) + (10^{12} - 25)) \]

3. **Simplify the Terms**:
- The first term simplifies to:
\[ (10^{12} + 25) - (10^{12} - 25) = 25 + 25 = 50 \]
- The second term simplifies to:
\[ (10^{12} + 25) + (10^{12} - 25) = 10^{12} + 25 + 10^{12} - 25 = 2 \cdot 10^{12} \]

4. **Combine the Results**:
Now, substituting back into the expression gives:
\[ 50 \cdot (2 \cdot 10^{12}) = 100 \cdot 10^{12} \]

5. **Express in Terms of \(10^n\)**:
We can rewrite \(100\) as \(10^2\):
\[ 100 \cdot 10^{12} = 10^2 \cdot 10^{12} = 10^{2 + 12} = 10^{14} \]

6. **Equate to Find \(n\)**:
We have \(10^{14} = 10^n\). Therefore, by comparing the exponents, we find:
\[ n = 14 \]

### Final Answer:
The value of \(n\) is \(14\).