Question

which multiplication expression is equivalent to \\(\frac{x + 8}{x^3} \div \frac{2x + 16}{2x^2}\\)?
\\(\frac{x + 8}{x^3} \cdot \frac{2x + 16}{2x^2}\\) \\(\frac{x^3}{x + 8} \cdot \frac{2x^2}{2x + 16}\\) \\(\frac{x + 8}{x^3} \cdot \frac{2x^2}{2x + 16}\\)

Explanation:

Step1: Recall Dividing Fractions Rule

To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction. The rule is \(\frac{a}{b} \div \frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}\), where \(b
eq0\), \(c
eq0\), \(d
eq0\).

Step2: Apply the Rule to the Given Expression

Given the expression \(\frac{x + 8}{x^{3}}\div\frac{2x + 16}{2x^{3}}\), here \(a=x + 8\), \(b=x^{3}\), \(c=2x + 16\), \(d=2x^{3}\).
Using the division of fractions rule, we rewrite the division as multiplication by the reciprocal of the second fraction. The reciprocal of \(\frac{2x + 16}{2x^{3}}\) is \(\frac{2x^{3}}{2x + 16}\).
So, \(\frac{x + 8}{x^{3}}\div\frac{2x + 16}{2x^{3}}=\frac{x + 8}{x^{3}}\cdot\frac{2x^{3}}{2x + 16}\).

Answer:

The third option: \(\boldsymbol{\frac{x + 8}{x^{3}}\cdot\frac{2x^{3}}{2x + 16}}\) (assuming the third option is \(\frac{x + 8}{x^{3}}\cdot\frac{2x^{3}}{2x + 16}\) as per the visual)