Just as you would add two numbers such as 2+3, you add two square roots in a similar fashion. The only difference between adding two square roots and adding two rational numbers (1,2,3…) is that you might not be able to simplify the answer down to a single number. Instead, you might be left with the most basic form, but still have a square root in there. When dealing with the addition of square roots, this is an entirely plausible and possible situation.
How to Add Square Roots
When you are adding square roots, the radicals (the square roots) have to be the same. That means that if you have sq (square root) of 2 and sq of 3, you cannot combine them together and put sq 5. However, if you have the scenario where you have (sq 9) + (sq 25), you can add those together. However, it would look like this:
(sq 9) + (sq 25) = 3 + 5 = 8
The reason for that is because both 9 and 25 are perfect squares. In other words, the square root of 9 is 3 and the square root of 25 is 5. Because you were able to break them apart like that, you could add them together and get the answer.
However, what happens if you have an equation like this:
2(sq 3) + 4(sq 3)
To solve this, you have to first pay attention to the fact that the radicals are the same (sq 3). Now that they are the same, you would simply combine the other two numbers. 2 + 4 is equal to six. Therefore, the problem would look like this:
2(sq 3) + 4(sq 3) = 6(sq 3)
Finally, what happens when you have an equation such as:
3(sq 8‌) + 5(sq 2)
They are obviously not the same radicals, so one might argue that you cannot add them. However, that’s not the case. You can break apart radicals into smaller forms so that you have a perfect square appear. For example:
(sq 8‌) = (sq 4)(sq 2)
Because we know that, we can now take the (sq 8‌) from the original problem and instead apply the (sq 4)(sq 2). This will make the equation looks like this:
3(sq 4)(sq 2) + 5(sq 2) = ?
If you’ve noticed, the (sq 4) is a perfect square. Therefore, the square root of 4 is 2. That means that the equation actually reads:
3×2(sq 2) + 5(sq 2) = 6(sq 2) + 5(sq 2) = 11(sq 2)
In other words, you were able to break apart the (sq 8‌) into a perfect square and a (sq 2) which meant that the radical was now equal to the opposite radical. With some basic arithmetic, the problem was easily solved by multiply the 3 and the 2 together and then adding the 6 to the 5.
The basic rule for adding square roots is that you are trying to simplify. Therefore, you want there to be no way for you to break the problem apart anymore. If you wound up with an answer such as 11(sq 4), that 4 can be broken up into 2 and the answer would actually be 22. You can only add like radicals and it is important to always simplify as much as possible.