Question

Can someone please explain in detail?

Answer 1

This definitely falls under the category of "things you don't really need to know"

You're trying to find two numbers that ADD up to the first number under large the radical and MULTIPLY to the number under the smaller radical.

So in their example, you need two numbers that add up to 8, and multiply to 15. That's 3 and 5, so you can rewrite the expression as √5 - √3.
You want to put the bigger number first, so the answer stays positive.

So for #1 you have 
___ + ___ = 7 and ___ x ____ = 10
So the two numbers are 5 and 2 and you can rewrite the expression as
√5 - √2

#2: ___ + ___ = 10 and ___ x ___ = 21
The numbers are 7 and 3 and the expression is
√7 - √3

#3: ___ + ___ = 15 and ___ x ___ = 50
The numbers are 5 and 10 and the expression is
√10 - √5

#4: ___ + ___ = 6 and ___ x ___ = 5
The numbers are 5 and 1 and the expression is
√5 - √1, which is √5 - 1