The square root of `(7 + 3 sqrt(5)) (7 - 3 sqrt(5))` is :

Options

A. 4
B. `sqrt(5)`
C. `3 sqrt(5)`
D. 2

Answer

To solve the problem, we need to find the square root of the expression \((7 + 3\sqrt{5})(7 - 3\sqrt{5})\).

### Step-by-Step Solution:

1. **Identify the Expression**:
We have the expression \((7 + 3\sqrt{5})(7 - 3\sqrt{5})\).

2. **Use the Difference of Squares Formula**:
The expression can be simplified using the difference of squares formula, which states that \( (a + b)(a - b) = a^2 - b^2 \). Here, let \( a = 7 \) and \( b = 3\sqrt{5} \).

3. **Calculate \( a^2 \)**:
\[ a^2 = 7^2 = 49 \]

4. **Calculate \( b^2 \)**:
\[ b^2 = (3\sqrt{5})^2 = 3^2 \cdot (\sqrt{5})^2 = 9 \cdot 5 = 45 \]

5. **Subtract \( b^2 \) from \( a^2 \)**:
\[ a^2 - b^2 = 49 - 45 = 4 \]

6. **Take the Square Root**:
Now, we need to find the square root of the result:
\[ \sqrt{4} = 2 \]

### Final Answer:
The square root of \((7 + 3\sqrt{5})(7 - 3\sqrt{5})\) is \(2\).

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