Question

1. the simplified form of the rational expression \\(\frac{-3a^2}{a - 5} \div \frac{-12a^5}{4a - 20}\\) is...

a. \\(\frac{9a^7}{(a - 5)^2}\\)
b. \\(\frac{1}{a^3}\\)
c. \\(\frac{2}{5}\\)
d. \\(a^3\\)

Explanation:

Step1: Rewrite division as multiplication

To divide rational expressions, we multiply by the reciprocal. So, \(\frac{-3a^2}{a - 5}\div\frac{-12a^5}{4a - 20}=\frac{-3a^2}{a - 5}\times\frac{4a - 20}{-12a^5}\)

Step2: Factor the denominator

Factor \(4a - 20\) as \(4(a - 5)\). Now the expression becomes \(\frac{-3a^2}{a - 5}\times\frac{4(a - 5)}{-12a^5}\)

Step3: Cancel common factors

Cancel out the common factors: \(-3\) and \(-12\) (leaving \(\frac{1}{4}\)), \(a^2\) and \(a^5\) (leaving \(\frac{1}{a^3}\)), and \(a - 5\) in the numerator and denominator. So we have \(\frac{1\times4\times1}{1\times4\times a^3}=\frac{1}{a^3}\)

Answer:

B. \(\frac{1}{a^3}\)