The square root of is equal to:

For an expression of the form a + b + 2√(ab), the identity (√a + √b)2 = a + b + 2√(ab) lets us 'denest' the nested surd: whenever the number under the outer root splits into two positive numbers whose sum equals its rational part and whose product equals the number under the inner root, the square root simplifies directly to √a + √b.

1. Match 7 + 2√10 to the form a + b + 2√(ab): here a + b = 7 and ab = 10.

2. Find two positive numbers with sum 7 and product 10 — these are 5 and 2, since 5 + 2 = 7 and 5 × 2 = 10.

3. Rewrite 7 + 2√10 as (√5)2 + (√2)2 + 2·√5·√2, which is exactly the expansion of (√5 + √2)2.

4. Taking the square root of both sides gives √(7 + 2√10) = √5 + √2, i.e. √2 + √5.

Squaring (√2 + √5) confirms the result: (√2 + √5)2 = 2 + 5 + 2√10 = 7 + 2√10, exactly the original expression — so √2 + √5 is indeed the required square root.