Question

algebra: concepts and connections - plc
multiplying and dividing rational expressions
if \\(\frac{a}{b}\\) and \\(\frac{x}{y}\\) represent rational expressions and \\(b \
eq 0\\) and \\(y \
eq 0\\), what is true of their product? choose three correct answers.
the product is a rational expression.
the product is the ratio of \\(ax\\) and \\(by\\).
the product is equivalent to \\(\frac{ax}{b}(y)\\).
the product is equivalent to \\(\frac{ay}{bx}\\).

Explanation:

Step1: Calculate product of rationals

The product of $\frac{a}{b}$ and $\frac{x}{y}$ is $\frac{a}{b} \times \frac{x}{y} = \frac{ax}{by}$.

Step2: Analyze each statement

1. "The product is a rational expression.": A rational expression is a ratio of polynomials. $ax$ and $by$ are polynomials, and $by

eq 0$ (since $b
eq 0, y
eq 0$), so this is true.

1. "The product is the ratio of $ax$ and $by$.": From Step1, the product is $\frac{ax}{by}$, which is the ratio of $ax$ to $by$, so this is true.
2. "The product is equivalent to $\frac{ax}{b}(y)$.": $\frac{ax}{b}(y) = \frac{axy}{b}$, which is not equal to $\frac{ax}{by}$, so this is false.
3. "The product is equivalent to $\frac{ay}{bx}$.": $\frac{ay}{bx}$ is not equal to $\frac{ax}{by}$ (unless $x^2=y^2$), so this is false.

Answer:

• The product is a rational expression.
• The product is the ratio of $ax$ and $by$.