Examine whether the following numbers are rational or irrational :
` (7- sqrt7) (7 + sqrt( 7))`

Answer

To determine whether the expression \((7 - \sqrt{7})(7 + \sqrt{7})\) is rational or irrational, we can follow these steps:

### Step-by-Step Solution:

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

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 \( b = \sqrt{7} \).

3. **Calculate \( a^2 \) and \( b^2 \)**:
- Calculate \( a^2 = 7^2 = 49 \).
- Calculate \( b^2 = (\sqrt{7})^2 = 7 \).

4. **Subtract \( b^2 \) from \( a^2 \)**:
\[ a^2 - b^2 = 49 - 7 = 42. \]

5. **Conclusion**: The result of the multiplication is \( 42 \). Since \( 42 \) is a whole number, it is a rational number.

### Final Answer:
Thus, \((7 - \sqrt{7})(7 + \sqrt{7})\) is a rational number.

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