# Adding and Subtracting Radicals

Square root radicals can be added or subtracted by combining like radicals, which are radicals with the same radicand. To do this, add or subtract their coefficients and keep the radicand the same. You can also simplify radicals if they are not like radicals.

## Adding and Subtracting Radicals

Square root radicals can be added or subtracted by combining like radicals. Like radicals are radicals with the same radicand, or number under the radical. To combine like radicals add or subtract their coefficients and keep the radicand the same.

Let’s practice on 2 square roots of 3, plus the square root of 3. These are like radicals since their radicands are the same, they’re both 3, so now we just need to add their coefficients and keep their radicands the same.

The coefficient of 2 square roots of 3 is 2, and the coefficient of the square root of 3 would be 1, since 1 times the square root of 3 is the square root of 3, so 2 plus 1 is 3 square roots of 3. These next 2 problems don’t have like radicals, but what we’re going to do is simplify them to see if maybe they do have like radicals when they’re simplified.

We’ll start by simplifying the square root of 32. 32 is not a perfect square, but it does have a perfect square factor, so we’re going to rewrite it as the square root of 16 times 2, since the perfect square factor of 32 is 16, plus, and 8 is also not a perfect square factor, but it has a perfect square factor of 4. We’re going to rewrite that as the square root of 4 times 2, since 4 times 2 is 8.

Now we can rewrite both of these using our rule, the square root of A times B is the square root of A times the square root of B. I am going to rewrite this as the square root of 16 times the square root of 2, plus the square root of 4, times the square root of 2. Again this is just the square root of 32, and this is the square root of 8, so we’re going to simplify that.

The square root of 16 is 4, since 4 times 4 is 16, 4 times the square root of 2 is 4 square roots of 2, plus the square root of 4 is 2 since 2 times 2 is 4, and 2 times the square root of 2 is 2 square roots of 2. Now that we’ve simplified the square root of 32 and the square root of 8, we see that they do have like radicals, or they are like radicals, because their radicands are the same, 2 and 2.

Now that they’re like radicals, we can combine them by adding their coefficients and keeping the radicand the same. 4 plus 2 is 6 square roots of 2. Let’s look at 1 more. Again we see that the radicals are not alike, since the radicands are not the same, 108 and 36. What we’re going to do is we’re going to have to simplify these to see if we could create like radicals.

108 is kind of a big number, and if you didn’t know what its factors were or what its perfect square factor was then you could factor tree it. I’m going to go over here, and I’m going to factor tree 108. I know right off the bat, since it ends in 8 (an even number) that I can divide it by 2.

2 times 54 would be 108, and 54 can be factor treed to 9 times 6, and 9 is a perfect square factor, (so I’m going to leave that alone) and I’m going to factor tree 6 to 2 times 3. Well I see here that I have 2 and 2, and 2 times 2 is 4, that’s also a perfect square factor.

Then if I take that 4 and I multiply it times 9, I get 36 which is also a perfect square factor. I could write 108 many different ways, I could write it as 2 times 2, times 3, times- and 9 would be 3 times 3, or I could do 2 times 2, times 3, times 9. I could write it as 4 times 9, times 3, but the quickest way would just be to write our perfect square factor of 36, times 3.

Again, that 36 came from multiplying 2 times 2 which is 4, times 9, which gives us 36. I’m going to rewrite the square root of 108, as the square root of 36 times 3, minus the square root of 36- 36 is a perfect square factor- so that’s just 6. Then I’m going to use my rule again on the square root of 36 times 3, to break it up and take the square root of 36.

I get the square root of 36, times the square root of 3, minus 6. This square root of 36, again 36 is a perfect square factor, so the square root of 36 is 6, since 6 times 6 is 36, and 6 times the square to 3 is 6 square roots of 3, then we bring down our minus 6. These are not like radicals, since this has a radical and this term does not, which means that I cannot simplify any further. So this would be the simplified form of the square root of 108 minus the square root of 36.