where a and b are positive integers, then the value of ab is closest to:

Given:
√(12 − 2√35) + √(8 + 2√15) = √a + √b
We simplify each surd separately.
Step 1: Simplify √(12 − 2√35)
Assume:
√(12 − 2√35) = √m − √n
Squaring both sides:
12 − 2√35 = m + n − 2√mn
Comparing terms:
m + n = 12
mn = 35
The numbers satisfying these conditions are:
m = 7 and n = 5
Therefore,
√(12 − 2√35) = √7 − √5
Step 2: Simplify √(8 + 2√15)
Assume:
√(8 + 2√15) = √p + √q
Squaring both sides:
8 + 2√15 = p + q + 2√pq
Comparing terms:
p + q = 8
pq = 15
The numbers are:
p = 5 and q = 3
Therefore,
√(8 + 2√15) = √5 + √3
Step 3: Add the expressions
(√7 − √5) + (√5 + √3)
= √7 + √3
So,
a = 7 and b = 3
Therefore,
ab = 7 × 3 = 21