1. **State the problem:** Given that $a^2 - b^2 = 45$ and $a - b = 5$, find the value of $\sqrt{a + b}$. 2. **Recall the formula:** The expression $a^2 - b^2$ can be factored using the difference of squares formula: $$a^2 - b^2 = (a - b)(a + b)$$ 3. **Substitute known values:** We know $a - b = 5$ and $a^2 - b^2 = 45$, so: $$45 = 5 \times (a + b)$$ 4. **Solve for $a + b$:** $$a + b = \frac{45}{5} = 9$$ 5. **Find $\sqrt{a + b}$:** $$\sqrt{a + b} = \sqrt{9} = 3$$ 6. **Interpret the result:** Since the square root function typically denotes the principal (non-negative) root, the value is $3$. **Final answer:** $\boxed{3}$