The square root of ((10 + √25)(12 – √49)) is:

Concept: To simplify the square root of a product like √[(a)(b)], first evaluate any perfect-square roots inside each factor, then multiply the simplified factors, and finally extract the largest perfect-square factor from the result using the identity √(m²·n) = m√n.

Applying it here:

1. Evaluate the inner square roots: √25 = 5 and √49 = 7.

2. Substitute back into each factor: (10 + 5) = 15 and (12 − 7) = 5.

3. Multiply the two simplified factors: 15 × 5 = 75.

4. Factor 75 to isolate the largest perfect square: 75 = 25 × 3.

5. Apply √(m²·n) = m√n: √75 = √(25 × 3) = 5√3.

Cross-check: Squaring 5√3 gives 25 × 3 = 75, which matches the product computed above, confirming the simplification.