find the sum of 4 radical 2 and 2 radical 8 in simplest form. Also, determine whether the result is rational or irrational and explain your answer.

To find the sum of \( 4\sqrt{2} \) and \( 2\sqrt{8} \), we first simplify \( 2\sqrt{8} \).

Since \( \sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4}\sqrt{2} = 2\sqrt{2} \), we can rewrite \( 2\sqrt{8} \) as:

\[ 2\sqrt{8} = 2 \cdot 2\sqrt{2} = 4\sqrt{2} \]

Now we can add \( 4\sqrt{2} \) and \( 4\sqrt{2} \):

\[ 4\sqrt{2} + 4\sqrt{2} = 8\sqrt{2} \]

Next, we determine whether \( 8\sqrt{2} \) is rational or irrational.

The number \( \sqrt{2} \) is known to be irrational, and multiplying an irrational number by a non-zero rational number (like 8) will result in an irrational number. Therefore, \( 8\sqrt{2} \) is irrational.

Now, to fill in the blanks in your statement:

The result is \( 8\sqrt{2} \), because it cannot be written as the ratio of two integers and its decimal expansion does not terminate or repeat.

So the filled statement is:

The result is \( 8\sqrt{2} \), because it cannot be written as the ratio of two integers and its decimal expansion does not terminate or repeat.

A. irrational or rational
B. can or cannot
C. does or does not

The expression \( 8\sqrt{2} \) is irrational.

Here's why:

• The number \( \sqrt{2} \) is known to be irrational, meaning it cannot be expressed as a fraction of two integers.
• When you multiply an irrational number (such as \( \sqrt{2} \)) by a non-zero rational number (like 8), the result remains irrational.

Thus, \( 8\sqrt{2} \) is irrational.