Question
Find the product of 5, square root of, 18, end square root5
18
and 5, square root of, 18, end square root5
18
in simplest form. Also, determine whether the result is rational or irrational and explain your answer.
18
and 5, square root of, 18, end square root5
18
in simplest form. Also, determine whether the result is rational or irrational and explain your answer.
Answers
GPT-4o mini
To find the product of \( 5\sqrt{18} \) and \( 5\sqrt{18} \), we can express it as:
\[
(5\sqrt{18}) \times (5\sqrt{18})
\]
This simplifies to:
\[
5 \times 5 \times \sqrt{18} \times \sqrt{18} = 25 \times (\sqrt{18})^2
\]
Since \((\sqrt{18})^2\) is simply \(18\), we can rewrite the expression as:
\[
25 \times 18
\]
Now, calculating this gives:
\[
25 \times 18 = 450
\]
Next, we can conclude whether \(450\) is rational or irrational. A number is considered rational if it can be expressed as the quotient of two integers. Since \(450\) is an integer, it is clearly rational.
Thus, the final product is:
\[
\boxed{450}
\]
and the result is rational since \(450\) is an integer.
\[
(5\sqrt{18}) \times (5\sqrt{18})
\]
This simplifies to:
\[
5 \times 5 \times \sqrt{18} \times \sqrt{18} = 25 \times (\sqrt{18})^2
\]
Since \((\sqrt{18})^2\) is simply \(18\), we can rewrite the expression as:
\[
25 \times 18
\]
Now, calculating this gives:
\[
25 \times 18 = 450
\]
Next, we can conclude whether \(450\) is rational or irrational. A number is considered rational if it can be expressed as the quotient of two integers. Since \(450\) is an integer, it is clearly rational.
Thus, the final product is:
\[
\boxed{450}
\]
and the result is rational since \(450\) is an integer.