Prove that the product of the two rational numbers 36−− [Invalid pasted math] and [Invalid pasted math] 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 two rational numbers is a rational number, we can start with the general definition of rational numbers. A rational number is defined as a number that can be expressed as the quotient of two integers \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \).

Let's represent the two rational numbers as:

\[ r_1 = \frac{a}{b} \] \[ r_2 = \frac{c}{d} \]

Here, \( a, b, c, \) and \( d \) are all integers, and \( b \) and \( d \) are not zero.

The product of \( r_1 \) and \( r_2 \) is:

\[ r_1 \times r_2 = \frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d} \]

Since the product \( a \cdot c \) and \( b \cdot d \) are both integers (as the product of integers is an integer), and \( b \cdot d \) is not zero (assuming \( b \) and \( d \) are not zero), the result \( \frac{a \cdot c}{b \cdot d} \) is also a rational number.

To summarize: the product of two rational numbers is a rational number because it can be expressed as a ratio of two integers.