1. **State the problem:** Simplify the expression $$\left(\frac{64m^{1/2}r^3}{m^{3/2}r^{1/4}}\right)^{1/2}$$ and find which of the given options is equivalent. 2. **Recall the rules:** - When dividing powers with the same base, subtract exponents: $$a^x / a^y = a^{x-y}$$ - When raising a power to another power, multiply exponents: $$(a^x)^y = a^{xy}$$ - Simplify inside the parentheses first, then apply the outer exponent. 3. **Simplify inside the parentheses:** $$\frac{64m^{1/2}r^3}{m^{3/2}r^{1/4}} = 64 \times m^{1/2 - 3/2} \times r^{3 - 1/4} = 64 \times m^{-1} \times r^{\frac{12}{4} - \frac{1}{4}} = 64 m^{-1} r^{\frac{11}{4}}$$ 4. **Apply the outer exponent $1/2$:** $$\left(64 m^{-1} r^{\frac{11}{4}}\right)^{1/2} = 64^{1/2} \times (m^{-1})^{1/2} \times \left(r^{\frac{11}{4}}\right)^{1/2} = 8 \times m^{-\frac{1}{2}} \times r^{\frac{11}{8}}$$ 5. **Rewrite with positive exponents:** $$8 r^{\frac{11}{8}} m^{-\frac{1}{2}} = \frac{8 r^{\frac{11}{8}}}{m^{\frac{1}{2}}}$$ 6. **Compare with options:** - None of the options exactly match this form. 7. **Check if options can be rewritten to match:** - Option A: $$\frac{8 r^{\frac{11}{4}}}{m^2}$$ exponent of $r$ is $\frac{11}{4} = 2.75$ which is not $\frac{11}{8} = 1.375$. - Option B: $$\frac{8 r^{\frac{11}{4}}}{m}$$ exponent of $r$ is still $2.75$. - Option C and D have coefficients 32, which is $64^{1/2} \times 2$, so no match. 8. **Re-examine the problem:** The original expression is raised to $1/2$, so the simplified form is $$8 r^{\frac{11}{8}} m^{-\frac{1}{2}}$$ which is not exactly any option. 9. **Possibility:** The problem might expect to simplify exponents differently or options might have a typo. 10. **Final answer:** None of the options exactly match the simplified expression $$\frac{8 r^{\frac{11}{8}}}{m^{\frac{1}{2}}}$$ but the closest is option B if the exponent of $r$ is considered approximate. **Answer:** B. $$\frac{8 r^{\frac{11}{4}}}{m}$$ (assuming a typo in the exponent of $r$ in the options).