Question

Find the product of $2 \sqrt{2}$ and $4 \sqrt{2}$ in simplest form. Also, determine whether the result is rational or irrational and explain your answer.

Step 1 ：The problem is asking for the product of two numbers, \(2 \sqrt{2}\) and \(4 \sqrt{2}\). To find the product, we simply multiply these two numbers together.

Step 2 ：The second part of the problem asks whether the result is rational or irrational. A rational number is a number that can be expressed as a fraction of two integers, while an irrational number cannot be expressed as a fraction.

Step 3 ：After calculating the product, we can determine whether it is rational or irrational by checking if it can be expressed as a fraction of two integers.

Step 4 ：The product of \(2 \sqrt{2}\) and \(4 \sqrt{2}\) is approximately 16.000000000000004, which is very close to 16. The slight discrepancy is likely due to the limitations of numerical precision.

Step 5 ：However, since the product is very close to 16, which is an integer, we can conclude that the product is a rational number.

Step 6 ：Final Answer: The product of \(2 \sqrt{2}\) and \(4 \sqrt{2}\) is \(\boxed{16}\), which is a rational number.