Find the product of 5, square root of, 10, end square root5 10 ​ and 2, square root of, 10, end square root2 10 ​ in simplest form.

Answer 1

To find the product of \(5\sqrt{10}\) and \(2\sqrt{10}\), we follow these steps:

1. Multiply the coefficients:

   • The coefficients are \(5\) and \(2\). \[ 5 \times 2 = 10 \]

2. Multiply the square roots:

   • We have \(\sqrt{10}\) and \(\sqrt{10}\). \[ \sqrt{10} \times \sqrt{10} = \sqrt{10^2} = 10 \]

3. Combine the results:

   • Now, we multiply the results from steps 1 and 2. \[ 10 \times 10 = 100 \]

Therefore, the product of \(5\sqrt{10}\) and \(2\sqrt{10}\) in simplest form is:

\[ \boxed{100} \]

Answer 2

The result is irrational because it CANNOT be written as the ratio of two integers and its decimal expansion does terminate or repeat. correct?

Answer 3

Actually, the statement about the result being irrational is incorrect in this context.

The product we calculated, \(100\), is a rational number because it can be expressed as a ratio of two integers:

\[ 100 = \frac{100}{1} \]

A number is considered irrational if it cannot be expressed as a fraction of two integers, and its decimal representation is non-terminating and non-repeating. Examples of irrational numbers include \(\pi\) and \(\sqrt{2}\).

In this case, since we found that the product \(100\) is a whole number (an integer), it is rational and can be written as the ratio of the integers \(100\) and \(1\), and its decimal expansion is \(100.0\), which does terminate.

So to clarify: the product \(100\) is a rational number, not irrational.