To prove that the product of the two rational numbers \(\sqrt{36}\) and \(\frac{5}{3}\) is a rational number, we first need to evaluate the square root of 36.
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Calculate \(\sqrt{36}\): \[ \sqrt{36} = 6 \]
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Express the numbers:
- \(\sqrt{36} = 6\), which is a rational number because it can be expressed as \(\frac{6}{1}\).
- \(\frac{5}{3}\) is already a rational number because it's in the form of \(\frac{p}{q}\) where \(p = 5\) and \(q = 3\) (both integers, and \(q \neq 0\)).
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Calculate the product: Now, we multiply the rational numbers: \[ 6 \times \frac{5}{3} \]
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Perform the multiplication: Multiply \(6\) by \(\frac{5}{3}\): \[ 6 \times \frac{5}{3} = \frac{6 \times 5}{3} = \frac{30}{3} = 10 \]
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Conclusion: The product is \(10\), which is a rational number.
We say a number is rational if it can be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\). Here, \(10\) can be expressed as \(\frac{10}{1}\), thus confirming it is a rational number.
Since both \(\sqrt{36}\) and \(\frac{5}{3}\) are rational numbers, their product is also a rational number. This is consistent with the property of rational numbers: the product of any two rational numbers is rational.
In summary:
- Product: \(10\)
- Type: Rational number (since \(10 = \frac{10}{1}\))
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