Question

multiplying and dividing rational expressions
what is the product?
\frac{a - 3}{15a} \cdot \frac{5}{a - 3}
options: \frac{1}{3a}, 3a, 3, \frac{1}{3}

Explanation:

Step1: Multiply numerators and denominators

Multiply the numerators: \((a - 3) \cdot 5 = 5(a - 3)\)
Multiply the denominators: \(15a \cdot (a - 3) = 15a(a - 3)\)
So the expression becomes \(\frac{5(a - 3)}{15a(a - 3)}\)

Step2: Cancel common factors

Cancel out the common factor \((a - 3)\) from numerator and denominator:
\(\frac{5\cancel{(a - 3)}}{15a\cancel{(a - 3)}} = \frac{5}{15a}\)

Step3: Simplify the fraction

Simplify \(\frac{5}{15a}\) by dividing numerator and denominator by 5:
\(\frac{5 \div 5}{15a \div 5} = \frac{1}{3a}\)

Answer:

\(\frac{1}{3a}\) (corresponding to the first option box with \(\frac{1}{3a}\))