1. **State the problem:** Simplify the expression $\sqrt{\frac{25}{4} + 16}$.\n\n2. **Recall the formula:** The square root of a sum is not the sum of the square roots, so we first simplify inside the square root.\n\n3. **Simplify inside the square root:** Convert 16 to a fraction with denominator 4 to combine easily: $16 = \frac{64}{4}$.\n\n4. **Add the fractions:** $\frac{25}{4} + \frac{64}{4} = \frac{25 + 64}{4} = \frac{89}{4}$.\n\n5. **Rewrite the expression:** $\sqrt{\frac{89}{4}} = \frac{\sqrt{89}}{\sqrt{4}} = \frac{\sqrt{89}}{2}$.\n\n6. **Final answer:** $\frac{\sqrt{89}}{2}$, which is the simplified form of the original expression.