1. **State the problem:** Calculate $\left(7 \frac{5}{12} - 5 \frac{3}{4}\right) \div 2^{2 \frac{1}{2}}$. 2. **Convert mixed numbers to improper fractions:** $7 \frac{5}{12} = \frac{7 \times 12 + 5}{12} = \frac{84 + 5}{12} = \frac{89}{12}$ $5 \frac{3}{4} = \frac{5 \times 4 + 3}{4} = \frac{20 + 3}{4} = \frac{23}{4}$ 3. **Find the difference:** $$\frac{89}{12} - \frac{23}{4} = \frac{89}{12} - \frac{23 \times 3}{12} = \frac{89}{12} - \frac{69}{12} = \frac{89 - 69}{12} = \frac{20}{12}$$ Simplify $\frac{20}{12}$: $$\frac{\cancel{20}}{\cancel{12}} = \frac{5}{3}$$ 4. **Calculate the divisor:** $2^{2 \frac{1}{2}} = 2^{\frac{5}{2}} = 2^{2 + \frac{1}{2}} = 2^2 \times 2^{\frac{1}{2}} = 4 \times \sqrt{2} = 4\sqrt{2}$ 5. **Divide the difference by the divisor:** $$\frac{5}{3} \div 4\sqrt{2} = \frac{5}{3} \times \frac{1}{4\sqrt{2}} = \frac{5}{3} \times \frac{1}{4\sqrt{2}} = \frac{5}{12\sqrt{2}}$$ 6. **Rationalize the denominator:** $$\frac{5}{12\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{5\sqrt{2}}{12 \times 2} = \frac{5\sqrt{2}}{24}$$ **Final answer:** $$\boxed{\frac{5\sqrt{2}}{24}}$$