The square root of a number is usually introduced via an activity that involves getting the side of a square with the given area. For example the side of a square with area 25 sq unit is 5 unit because 5 x 5 = 25. To introduce the existence of $\sqrt{5}$, a square of area 5 sq units is shown. The task is to find the length of its side. The student measures it then square the measure to check if it will equal to 5. Of course it won’t so they will keep on adjusting it. The teacher then introduces the concept of getting the root and the symbol used. This is a little boring.  A more challenging task is to start with this problem: Construct a square which is double the area of a given square. In my post GeoGebra and Mathematics: Squares and Square Roots I described a teaching sequence for introducing the idea of square root using this problem. There are 4 activities in the sequence. The construction below can be an extension of Activity 4. This extension can be used to teach simplifying radicals and addition of radicals. The investigation still uses the regular polygon tool  and introduces the text tool of GeoGebra.  Click links for the tutorial on how to use these tools. You will find the procedure for constructing the figure here.

The construction shows the following equivalence:

1. $2\sqrt{5} = \sqrt{5}+\sqrt{5}$ since EA = EF+FH

2. $4\sqrt{5} = 2\sqrt{5}+2\sqrt{5}$ since AK = AB+BJ

3. $2\sqrt{10} =\sqrt{10}+\sqrt{10}$

4. $4\sqrt{10} = 2\sqrt{10}+2\sqrt{10}$

5.$7\sqrt{5} = \sqrt{5}+2\sqrt{5}+4\sqrt{5}$

6. $2\sqrt{5} = \sqrt{20}$ because they are both lengths of the sides of square EHBA or poly3 whose area is 20 (see algebra panel)

7. $2\sqrt{10} = \sqrt{40}$ because they are both lengths of the sides of square AHJI or poly4 whose area is 40.

8. $4\sqrt{5} = \sqrt{80}$ because they are both sides of square AJLK or poly5 whose area is 80.