Find the product of 3 V5 and 2 V2 in simplest form. Also, determine whether the result is rational or irrational and explain your answer.

Answer 1

To find the product of \( 3 \sqrt{5} \) and \( 2 \sqrt{2} \), we follow these steps:

1. Multiply the coefficients (the numerical parts): \[ 3 \times 2 = 6 \]

2. Multiply the square roots: \[ \sqrt{5} \times \sqrt{2} = \sqrt{10} \]

Now we combine both results: \[ 3 \sqrt{5} \times 2 \sqrt{2} = 6 \sqrt{10} \]

The simplest form of the product is therefore: \[ 6 \sqrt{10} \]

Next, we need to determine whether \( 6 \sqrt{10} \) is a rational or irrational number.

• The number \( 6 \) is a rational number because it can be expressed as the fraction \( \frac{6}{1} \).
• The square root \( \sqrt{10} \) is irrational, as 10 is not a perfect square.

The product of a rational number (6) and an irrational number (\( \sqrt{10} \)) is always irrational.

Thus, the final conclusion is that: \[ 6 \sqrt{10} \text{ is irrational.} \]