What is the square root of (8 + 2√15)?

Concept: To simplify a nested radical of the form √(m + 2√n), find two positive numbers p and q such that p + q = m and pq = n. Then m + 2√n = p + q + 2√(pq) = (√p + √q)², so √(m + 2√n) = √p + √q.

Application:

1. Here m = 8 and n = 15, so we need p and q with p + q = 8 and pq = 15.

2. p = 5 and q = 3 work, since 5 + 3 = 8 and 5 × 3 = 15.

3. So 8 + 2√15 = 5 + 3 + 2√5·√3 = (√5)² + (√3)² + 2(√5)(√3) = (√5 + √3)².

4. Taking the square root of both sides: √(8 + 2√15) = √5 + √3.

Cross-check: (√5 + √3)² = 5 + 3 + 2√15 = 8 + 2√15, which matches the original expression, confirming the result.