Prove that the product of the two rational numbers 36−−√

Question

and 53
is a rational number. Find the product and explain how you know it is a rational number. Explain your reasoning in 1–2 sentences.(2 points)

Answer 1

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

We know: \[ \sqrt{36} = 6 \] So, the product is: \[ 6 \times \frac{5}{3} = \frac{6 \times 5}{3} = \frac{30}{3} = 10 \]

Now, since both \( 6 \) and \( \frac{5}{3} \) are rational numbers (because \( 6 \) can be expressed as \( \frac{6}{1} \) and \( \frac{5}{3} \) is already in rational form), their product is also a rational number. In general, the product of any two rational numbers is rational because the product of two fractions (or integers) results in another fraction (or integer).

Thus, the product \( 10 \) is a rational number.