find the product of radical 7 and 2 radical 2 in simplest form. Also, determine whether the result is rational or irrational and explain your answer.

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.