Question

q7 divide the rational expressions and simplify.
\\(\frac{2q}{p^3} \div \frac{4q}{-p}\\)
\\(\boldsymbol{\circ} -\frac{1}{2p^2}\\)
\\(\boldsymbol{\circ} -\frac{1}{2p}\\)
\\(\boldsymbol{\circ} -\frac{8q^2}{p^4}\\)
\\(\boldsymbol{\circ} -\frac{p^4}{8q^2}\\)

Explanation:

Step1: Recall division of rational expressions

To divide two rational expressions, we multiply the first by the reciprocal of the second. So, $\frac{2q}{p^3} \div \frac{4q}{-p} = \frac{2q}{p^3} \times \frac{-p}{4q}$.

Step2: Multiply numerators and denominators

Multiply the numerators: $2q \times (-p) = -2pq$. Multiply the denominators: $p^3 \times 4q = 4p^3q$. So we have $\frac{-2pq}{4p^3q}$.

Step3: Simplify the fraction

Cancel out common factors. The $q$ terms cancel, and we can simplify the coefficients and $p$ terms. $\frac{-2}{4} = -\frac{1}{2}$, and $\frac{p}{p^3} = \frac{1}{p^2}$. So combining these, we get $-\frac{1}{2p^2}$.

Answer:

$-\frac{1}{2p^2}$ (corresponding to the first option, e.g., A. $-\frac{1}{2p^2}$ if the first option is labeled A)