Question

simplify the rational expression.\\(\frac{4u^{2}v - 12uv^{2}}{18uv}\\)

Explanation:

Step1: Factor numerator and denominator

First, factor out the greatest common factor (GCF) from the numerator. The GCF of \(4u^{2}v\) and \(-12uv^{2}\) is \(4uv\). So, the numerator becomes \(4uv(u - 3v)\). The denominator is \(18uv\), and we can factor out \(2uv\) (or just keep it as is for now and look for common factors).
\[
\frac{4uv(u - 3v)}{18uv}
\]

Step2: Cancel common factors

Now, we can cancel out the common factors in the numerator and the denominator. The common factors are \(uv\) (since \(uv\) appears in both the numerator and the denominator). Also, we can simplify the coefficients \(4\) and \(18\) by dividing both by their greatest common divisor, which is \(2\). So, \(4\div2 = 2\) and \(18\div2=9\).
After canceling \(uv\) and simplifying the coefficients, we get:
\[
\frac{2(u - 3v)}{9}
\]
Or we can expand the numerator to get \(\frac{2u-6v}{9}\)

Answer:

\(\frac{2u - 6v}{9}\) (or \(\frac{2(u - 3v)}{9}\))