√[ 7 + 2√10 ]
Let us suppose that we can write this in a final form of √a + √b
Square both sides
7 + 2√10 = a + 2√[ab] + b
Equating coefficients we have that
a + b = 7
2√10 = 2√[ab]
Then ab = 10 ⇒ b = 10 / a
And
a + 10/a = 7 multiply through by a
a^2 + 10 = 7a rearrange as
a^2 - 7a + 10 = 0 factor
(a - 5) ( a - 2) = 0
Setting each factor to 0 and solving for a we get that
a = 5 a = 2
When a = 5, b = 10/5 = 2
or
When a = 2, b = 10/2 = 5
So.....depending on your preference for a , b.......we have
√[ 7 + 2√10 ] = √2 + √5