Question

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.

Answer 1

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.

1. **Calculate \(\sqrt{36}\)**:
\[
\sqrt{36} = 6
\]

2. **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 \(q \neq 0\)).

3. **Calculate the product**:
Now, we multiply the rational numbers:
\[
6 \times \frac{5}{3}
\]

4. **Perform the multiplication**:
Multiply \(6\) by \(\frac{5}{3}\):
\[
6 \times \frac{5}{3} = \frac{6 \times 5}{3} = \frac{30}{3} = 10
\]

5. **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 \(10 = \frac{10}{1}\))