Question

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.

Result: ?

the result is (irrational or rational) because it (can or cannot) be written as the ratio of two integers and its decimal expansion (does or does not) terminate or repeat.

Answer 1

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.