Question

question 38 of 42
if \\(\frac{a}{b}\\) and \\(\frac{c}{d}\\) are rational expressions, then which of the following is true?
\\(\bigcirc\\) a. \\(\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{b}{a} \bullet \frac{c}{d}\\)
\\(\bigcirc\\) b. \\(\frac{a}{b} \bullet \frac{c}{d} = \frac{a \cdot c}{b \cdot d}\\)
\\(\bigcirc\\) c. \\(\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \bullet c}{b \bullet d}\\)
\\(\bigcirc\\) d. \\(\frac{a}{b} \bullet \frac{c}{d} = \frac{a \bullet d}{b \bullet c}\\)

Explanation:

Step1: Recall the rule for multiplying rational expressions

The rule for multiplying two rational expressions \(\frac{A}{B}\) and \(\frac{C}{D}\) (where \(B
eq0\) and \(D
eq0\)) is that we multiply the numerators together and the denominators together. So, \(\frac{A}{B}\cdot\frac{C}{D}=\frac{A\times C}{B\times D}\).

Step2: Analyze each option

• Option A: Let's simplify the right - hand side \(\frac{B}{A}\cdot\frac{C}{D}=\frac{B\times C}{A\times D}\), and the left - hand side is \(\frac{\frac{A}{B}}{\frac{C}{D}}=\frac{A}{B}\times\frac{D}{C}=\frac{A\times D}{B\times C}\). These two are not equal, so option A is incorrect.
• Option B: According to the rule of multiplying rational expressions, \(\frac{A}{B}\cdot\frac{C}{D}=\frac{A\times C}{B\times D}\), so this option is correct.
• Option C: The left - hand side \(\frac{\frac{A}{B}}{\frac{C}{D}}=\frac{A}{B}\times\frac{D}{C}=\frac{A\times D}{B\times C}\), and the right - hand side is \(\frac{A\times C}{B\times D}\). These two are not equal, so option C is incorrect.
• Option D: The left - hand side \(\frac{A}{B}\cdot\frac{C}{D}=\frac{A\times C}{B\times D}\), and the right - hand side is \(\frac{A\times D}{B\times C}\). These two are not equal, so option D is incorrect.

Answer:

B. \(\frac{A}{B}\cdot\frac{C}{D}=\frac{A\cdot C}{B\cdot D}\)