prove that the product of the two rational numbers 36squared and 5/3 is a rational numebr. find the product and explain how you know it is a rational number in 1 sentence

Answer 1

To prove that the product of the two rational numbers \(36^2\) and \(\frac{5}{3}\) is a rational number, we first calculate the product:

\[ 36^2 = 1296 \]

Now, we find the product:

\[ 1296 \times \frac{5}{3} = \frac{1296 \times 5}{3} = \frac{6480}{3} = 2160 \]

Since both \(36^2\) and \(\frac{5}{3}\) can be expressed as the ratio of integers (\(36^2\) as \(1296/1\) and \(\frac{5}{3}\) as \(5/3\)), their product \(2160\) is also an integer, and thus it is a rational number. Consequently, any product of two rational numbers remains rational, exemplified here by \(2160\).