To determine which expression is equivalent to \(\frac{10 r^2 x^3}{8 r g^2}\), we need to simplify this original expression step-by-step. Here's the detailed approach:
1. Simplify the Coefficient:
[tex]\[
\frac{10}{8} = \frac{5}{4}
\][/tex]
2. Simplify the Powers of \(r\):
[tex]\[
\frac{r^2}{r^1} = r^{2-1} = r^1 = r
\][/tex]
3. Resulting Simplified Expression:
Combining the simplified coefficient and the simplified power of \(r\), the expression \(\frac{10 r^2 x^3}{8 r g^2}\) simplifies to:
[tex]\[
\frac{5}{4} \cdot r \cdot \frac{x^3}{g^2}
\][/tex]
Given the provided answers, none directly represent this simplified expression. But evaluating each option for equivalence:
A. \(\frac{2 c^3}{1}\) — This does not involve \(r, x\), or \(g\), and hence cannot be equivalent to the original expression.
B. \(\frac{2 r^4}{s^1}\) — This involves different powers and variables, so it is not equivalent.
C. \(\frac{8 r^4}{r}\) — Simplifying this:
[tex]\[
\frac{8 r^4}{r} = 8 r^{4-1} = 8 r^3
\][/tex]
This does not match the form \(\frac{5}{4} r \cdot \frac{x^3}{g^2}\).
D. \(\frac{81}{s^1}\) — This does not involve \(r, x, \)or \(g\), so it cannot be equivalent to the original expression.
After evaluating all the options, none of the provided choices are equivalent to the simplified form [tex]\(\frac{5}{4} r \cdot \frac{x^3}{g^2}\)[/tex]. Therefore, the best answer is that none of the given options (A, B, C, or D) are equivalent to the original expression.
