\(10\sqrt{10}=(\sqrt{100})(\sqrt{10})=\sqrt{1000}\)

1000 = (2)(2)(2)(5)(5)(5)

Let's use the above prime factorization to see see how many different perfect squares we can factor out 1000.

For example, 1000 = 4 x 250 (where 4 is a perfect square)
So, \(\sqrt{1000} = \sqrt{4} \times \sqrt{250} = 2\sqrt{250}\)
In this case, \(a = 2\) and \(b = 250\), which means \(a + b = 2 + 250 = 252\)
So, answer choice B works.

Also, 1000 = 25 x 40 (where 25 is a perfect square)
So, \(\sqrt{1000} = \sqrt{25} \times \sqrt{40} = 5\sqrt{40}\)
In this case, \(a = 5\) and \(b = 40\), which means \(a + b = 5 + 40 = 45\)
So, answer choice A works.

What about answer choice C?
The only way to get CLOSE to a sum of 1000, is to factor out a 1 from 1000
In other words, let's examine 1000 = 1 x 1000 (where 1 is a perfect square)
Here, \(\sqrt{1000} = \sqrt{1} \times \sqrt{1000} = 1\sqrt{1000}\)
In this case, \(a = 1\) and \(b = 1000\), which means \(a + b = 1 + 1000 = 1001\)
Close, but no good.

Answer: A, B

Cheers,
Brent