If 𝐴𝐵BA​ and 𝐶𝐷DC​ are rational expressions, then 𝐴𝐵∙𝐶𝐷=𝐴∙𝐶𝐵∙𝐷BA​ ∙ DC​ = B∙DA∙C​ .A.TrueB.FalseSUBMITarrow_backPREVIOUS

To determine whether the statement $AB \cdot CD = A \cdot C \cdot \frac{B}{D}$ is true or false, let's analyze both sides of the equation.

1. Break Down the Problem

• We have two products of rational expressions: $AB$ and $CD$.
• We need to see if their product equals $A \cdot C \cdot \frac{B}{D}$.

2. Relevant Concepts

• The product of two fractions can be expressed as: $\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$
• Thus, for the rational expressions, the product $AB \cdot CD$ can be expanded as: $AB \cdot CD = \frac{A \cdot B \cdot C \cdot D}{1} = A \cdot C \cdot B \cdot D$

3. Analysis and Detail

• The left-hand side can be interpreted as: $AB \cdot CD = A \cdot B \cdot C \cdot D$
• The right-hand side, as stated: $A \cdot C \cdot \frac{B}{D}$ simplifies to: $A \cdot C \cdot B \cdot D^{-1}$
• Thus, we can rewrite this as: $A \cdot C \cdot B \cdot \frac{1}{D} = A \cdot C \cdot \frac{B}{D}$
• Hence, both sides cannot express the same rational expression unless $D = 1$ because one side includes $D$ while the other includes $\frac{1}{D}$.

4. Verify and Summarize

• When we equate both sides: $A \cdot B \cdot C \cdot D = A \cdot C \cdot \frac{B}{D} \cdot D$
• This equation does not hold for all $A, B, C, D$ when $D \neq 1$.

Final Answer

The statement is B. False.