1. **Stating the problem:** We are given the binomial coefficient expression $$\binom{r^3}{\frac{r}{t}}$$ and asked to find an equivalent expression. 2. **Understanding the binomial coefficient:** The binomial coefficient $$\binom{n}{k}$$ is defined for integers $n$ and $k$ as $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ where $n!$ is the factorial of $n$. 3. **Important rules:** - The binomial coefficient is typically defined for non-negative integers $n$ and $k$ with $k \leq n$. - When the inputs are algebraic expressions, the binomial coefficient can be interpreted using the Gamma function or generalized binomial theorem, but it is not simply a fraction or product. 4. **Analyzing the given expression:** Here, the numerator is $r^3$ and the denominator is $\frac{r}{t}$. 5. **Simplifying the denominator:** $$\frac{r}{t} = r \cdot \frac{1}{t}$$ 6. **Equivalent expressions:** The binomial coefficient $$\binom{r^3}{\frac{r}{t}}$$ cannot be simplified to a simple algebraic fraction or product without additional context or restrictions on $r$ and $t$. 7. **Conclusion:** The expression $$\binom{r^3}{\frac{r}{t}}$$ is already in its simplest binomial coefficient form and does not have a simpler equivalent expression unless $r$ and $t$ are specified as integers and factorials are defined. **Final answer:** The expression is equivalent to itself: $$\binom{r^3}{\frac{r}{t}}$$