If `a ** b = a ^(2) + b ^(2) and a . b = a ^(2) - b ^(2),` then the value of (5 * 2) . 25 is

Options

A. 225
B. 226
C. 216
D. 215

Answer

To solve the problem, we need to evaluate the expression \( (5 * 2) . 25 \) using the definitions provided for the operations \( * \) and \( . \).

### Step-by-Step Solution:

1. **Identify the operations**:
- The operation \( a * b \) is defined as \( a^2 + b^2 \).
- The operation \( a . b \) is defined as \( a^2 - b^2 \).

2. **Calculate \( 5 * 2 \)**:
- Using the definition of \( * \):
\[ 5 * 2 = 5^2 + 2^2 \]
- Calculate \( 5^2 \) and \( 2^2 \):
\[ 5^2 = 25 \quad \text{and} \quad 2^2 = 4 \]
- Now add these results:
\[ 5 * 2 = 25 + 4 = 29 \]

3. **Now calculate \( 29 . 25 \)**:
- Using the definition of \( . \):
\[ 29 . 25 = 29^2 - 25^2 \]
- Calculate \( 29^2 \) and \( 25^2 \):
\[ 29^2 = 841 \quad \text{and} \quad 25^2 = 625 \]
- Now subtract these results:
\[ 29 . 25 = 841 - 625 = 216 \]

4. **Final Answer**:
- The value of \( (5 * 2) . 25 \) is \( 216 \).