Prove that the product of the two rational numbers sqrt (36) and 5/3 is a rational number. Find the product and explain how you know it is a rational number. Explain your reasoning in 1-2 sentences.

Answer 1

To find the product of the two rational numbers \(\sqrt{36}\) and \(\frac{5}{3}\), we first calculate \(\sqrt{36}\):

\[ \sqrt{36} = 6 \]

Now we multiply this by \(\frac{5}{3}\):

\[ 6 \times \frac{5}{3} = \frac{6 \times 5}{3} = \frac{30}{3} = 10 \]

The result is \(10\), which is a rational number since it can be expressed as \(\frac{10}{1}\).

Rational numbers are defined as numbers that can be expressed in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\). Since both \(\sqrt{36} = 6\) and \(\frac{5}{3}\) are rational, and the product of two rational numbers is also a rational number, this confirms that \(10\) is rational.