How might I have anticipated that $\frac14(\sqrt{5+2\sqrt5}+\sqrt{10+2\sqrt{5}})$ simplifies to a single surd (namely, $\frac14\sqrt{25+10\sqrt{5}}$)? This is perhaps a silly question related to calculating with surds. I was working out the area of a regular pentagon ABCDE of side length 1 today and I ended up with the following expression :
$$\frac{\sqrt{5+2\sqrt5}+\sqrt{10+2\sqrt{5}}}{4}$$
obtained by summing the areas of the triangles ABC, ACD and ADE.
I checked my solution with Wolfram Alpha which gave me the following equivalent expression :
$$\frac{\sqrt{25+10\sqrt{5}}}{4}$$
I was able to show that these two expressions are equivalent by squaring the numerator in my expression, which gave me
$$15+4\sqrt5+2\sqrt{70+30\sqrt5},$$
and then "noticing" that
$$\sqrt{70+30\sqrt5}=\sqrt{25+30\sqrt5+45}=5+3\sqrt5.$$
My question is the following : how could I have known beforehand that my sum of surds could be expressed as a single surd, and is there a way to systematize this type of calculation ? I would have liked to find the final, simplest expression on my own without the help of a computer.
Thanks in advance !

$10 + 2 \sqrt 5$   has norm $100 - 5 \cdot 4 = 80.$ $5 + 2 \sqrt 5$   has norm $25 - 5 \cdot 4 = 5.$   The ratio of the norms is $\frac{80}{5} = 16,$  which is an integer and a square, so the ratio might be very nice.
$$ \frac{10+2 \sqrt 5}{5 + 2 \sqrt 5} \cdot \frac{5-2 \sqrt 5}{5 - 2 \sqrt 5} 
=   \frac{30-10 \sqrt 5}{5 }  = 6 - 2 \sqrt 5  $$
Next, $36 - 5 \cdot 4 = 16$  so $ 6 - 2 \sqrt 5 $  might be a square.  Indeed, by inspection it is $\left( 1 - \sqrt 5  \right)^2 =  \left(  \sqrt 5  - 1  \right)^2$    So  $10 + 2 \sqrt 5  = \left(  \sqrt 5  - 1  \right)^2 \left( 5 + 2 \sqrt 5 \right) $   and
$  \sqrt{10 + 2 \sqrt 5}  =  \left(  \sqrt 5  - 1  \right) \sqrt { 5 + 2 \sqrt 5  } $   Thus
$$  \sqrt{10 + 2 \sqrt 5} + \sqrt { 5 + 2 \sqrt 5  } =  \left(  \sqrt 5    \right) \sqrt { 5 + 2 \sqrt 5  } =  \sqrt { 25 + 10 \sqrt 5  }$$
$$ \color{red}{ \sqrt{10 + 2 \sqrt 5} + \sqrt { 5 + 2 \sqrt 5  } =   \sqrt { 25 + 10 \sqrt 5  } } $$
