Adding radicals can only be accomplished if there are like terms within the radical. That means that the number inside the radical (square root) has the be the same as the number inside the other radical. If that is not the case, the two cannot be added together. If it is the case, though, the two can be added together.

Adding Radicals

For the sake of this guide, we’ll use the equation:

2*sq(20x) + 3*sq(5x)

Now, as can be seen from this equation, the numbers inside the radicals are different. The first is 20x and the second is 5x. The radicals can’t be added together yet, but the square root of 20x can be simplified. What this means is that 20x can be broken apart. This is what it look like:

2*sq(4*5x)

In other words, the 20x was broken apart into 4 and 5x. When you multiply 4 by 5x, you wind up with 20x. Therefore, you can break apart the 20x into something that is easier to work with. Now, this is advantageous because 4 is a perfect square. A perfect square is a number that is found when two of the same number are multiplied together. In the case of 4, 2×2 is equal to 4. Therefore, we can pull that 4 out of the square root and make it a 2 because the square root of 4 is 2. It’ll look like this:

2*sq(4)*sq(5x)

2*2*sq(5x)

4*sq(5x)

This is exactly what you want to see. If you remember the original equation, you know that to add the two radicals together, there needed to be like terms within the radical. Now that you’ve simplified, you’ve got those like terms. The equation looks like this:

4*sq(5x) + 3*sq(5x)

Since they are like terms, you can add. When adding radicals, the numbers within the square root don’t change. However, you do add the numbers outside of the square roots. So, with the above equation, the final answer that you’d have is:

7*sq(5x)

The four and the three, when added together, result in 7. The 5x is combined with the other 5x, but it stays as 5x. That’s why you want to simplify and get like terms. Adding the radicals becomes a piece of cake when all you need to do is add the numbers outside of the radicals.