To find the product of \( \sqrt{7} \) and \( 2\sqrt{2} \), we can proceed as follows:
\[ \sqrt{7} \times 2\sqrt{2} = 2(\sqrt{7} \times \sqrt{2}) = 2\sqrt{14} \]
Now, the simplified result is \( 2\sqrt{14} \).
Next, we need to determine if this result is rational or irrational.
- The number \( \sqrt{14} \) is irrational because 14 is not a perfect square. Therefore, it cannot be expressed as a fraction of two integers (i.e., it cannot be written as \( \frac{a}{b} \) where \( a \) and \( b \) are integers).
- Multiplying an irrational number \( \sqrt{14} \) by a rational number (2, in this case) will also yield an irrational number.
Thus, \( 2\sqrt{14} \) is also irrational.
Final answer:
Result: \( 2\sqrt{14} \)
The result is irrational because it cannot be written as the ratio of two integers, and its decimal expansion does not terminate or repeat.