To determine the product of the two rational numbers, we first calculate the product of \(\sqrt{36}\) and \(\frac{5}{3}\).
Step 1: Calculate \(\sqrt{36}\).
\[ \sqrt{36} = 6 \]
Step 2: Multiply \(6\) by \(\frac{5}{3}\):
\[ 6 \times \frac{5}{3} = \frac{6 \times 5}{3} = \frac{30}{3} = 10 \]
So, the product of \(\sqrt{36}\) and \(\frac{5}{3}\) is \(10\).
Explanation of Rationality
A number is considered rational if it can be expressed as the quotient of two integers (where the denominator is not zero).
In our case, the product \(10\) can be expressed as:
\[ \frac{10}{1} \]
where \(10\) and \(1\) are both integers and the denominator \(1\) is not zero. Therefore, \(10\) is a rational number.
Additionally, both \(\sqrt{36}\) and \(\frac{5}{3}\) are rational numbers:
- \(\sqrt{36} = 6\) is rational because \(6\) can be expressed as \(\frac{6}{1}\).
- \(\frac{5}{3}\) is rational by definition, as it is a fraction where both the numerator and denominator are integers.
When you multiply two rational numbers, the product is also always a rational number. Thus, the product of \(\sqrt{36}\) and \(\frac{5}{3}\) is indeed a rational number and is equal to \(10\).
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