To determine whether the statement "If [tex]\(\frac{A}{B}\)[/tex] and [tex]\(\frac{C}{D}\)[/tex] are rational expressions, then [tex]\(\frac{A}{B} \cdot \frac{C}{D} = \frac{A \cdot D}{B \cdot C}\)[/tex]" is true or false, we need to analyze how rational expressions are multiplied.
When multiplying two rational expressions, the rule is:
[tex]\[
\frac{A}{B} \cdot \frac{C}{D} = \frac{A \cdot C}{B \cdot D}
\][/tex]
Let’s demonstrate this rule with a general multiplication of rational expressions. Here’s a step-by-step breakdown:
1. Multiply the numerators: Take the numerator of the first fraction ([tex]\(A\)[/tex]) and multiply it by the numerator of the second fraction ([tex]\(C\)[/tex]).
[tex]\[
A \cdot C
\][/tex]
2. Multiply the denominators: Take the denominator of the first fraction ([tex]\(B\)[/tex]) and multiply it by the denominator of the second fraction ([tex]\(D\)[/tex]).
[tex]\[
B \cdot D
\][/tex]
3. Combine the fractions: Place the product of the numerators over the product of the denominators.
[tex]\[
\frac{A \cdot C}{B \cdot D}
\][/tex]
Thus, the correct multiplication of [tex]\(\frac{A}{B}\)[/tex] and [tex]\(\frac{C}{D}\)[/tex] results in [tex]\(\frac{A \cdot C}{B \cdot D}\)[/tex].
However, the given statement is:
[tex]\[
\frac{A}{B} \cdot \frac{C}{D} = \frac{A \cdot D}{B \cdot C}
\][/tex]
This statement claims that the product of the fractions results in [tex]\(\frac{A \cdot D}{B \cdot C}\)[/tex], which is incorrect based on the rules for multiplying rational expressions.
Therefore, the correct answer to the question is:
B. False
