1. **State the problem:** Simplify the expression $(5\sqrt{r})^2$ and compare it with the given expressions involving rational exponents of $r$. 2. **Recall the formula and rules:** - The square of a product: $(ab)^2 = a^2 b^2$. - The square root can be written as a rational exponent: $\sqrt{r} = r^{\frac{1}{2}}$. - When raising a power to another power: $(r^m)^n = r^{mn}$. 3. **Simplify $(5\sqrt{r})^2$ step-by-step:** $$(5\sqrt{r})^2 = 5^2 \times (\sqrt{r})^2$$ $$= 25 \times (r^{\frac{1}{2}})^2$$ $$= 25 \times r^{\frac{1}{2} \times 2}$$ $$= 25 \times r^1$$ $$= 25r$$ 4. **Compare with the given expressions:** - $r^{\frac{5}{2}}$ means $r^{2.5}$, which is not equal to $25r$. - $r^3$ means $r$ cubed, also not equal to $25r$. - $r^{\frac{2}{5}}$ means $r^{0.4}$, again not equal to $25r$. 5. **Conclusion:** The simplified form of $(5\sqrt{r})^2$ is $25r$, which is different from all the given rational exponent expressions. **Final answer:** $$(5\sqrt{r})^2 = 25r$$