Adding and Subtracting Radical Expressions - Problem 2

When I’m looking at this problem, it looks like I can’t do any simplifying because when I’m looking at these radicands, they all look totally different, but I could combine them if they were the same radicands, and you’ll see in problems often, these are the same radicands in disguise. Let’s go through and simplify each one of these.

4 square root of 5 is already simplified, I don’t need to do any work, but the square root of 45, I’m going to rewrite as two different numbers, the product of 2 numbers that I think has a perfect square like a number that goes into 45 is 9. I could rewrite this as 9 times 5 and I chose to use 9 and 5 because is a perfect square. So really this term when I include the -3 is -3 times regular 3 times 5, the square root of 5 excuse me, so this thing is really actually -9 root 5. Now I can see those guys are going to be able to be combined because they have the same radicands. Let’s come back to that.

Let’s look at this third piece here. We’ve got to simplify the square root of 80. Well, I want to look for numbers that multiply to 80 especially if one of them is a perfect square. The first one that comes to my head is 4 times 20, you might have had something different in your head, don’t worry we’ll get the same answer in the end. Well 10 times square root of 4 is like 10 times 2, right? So what I really have is 20 square root of 20, but I’m not done because 20 it’s under the radicand could be simplified even further, I could write that as square root of 4 times square root of 5. I used 4 and 5 because 4 is a perfect square. You guys might have though of 2 and 10, but neither 2 nor 10 is a perfect square so it’s not going to help you. So this piece is really going to be 20 times 2 root 5. 20 times 2 is 40, so I have 40 root 5.

Okay, now that these are simplified, let’s rewrite the original problem with all of my simplified radicands and then try to combine them together 4 root take away 9 root 5 and then plus 40 root 5 from this last piece. My final answer is going to come from combining these outside terms 4-9 is 5, -9 plus 40 is 35, so my final answer is going to be 35 square root of 5.

Who knew? This whole big messy thing with 3 chunkers becomes just 35 root 5. It’s a pretty cool problem I think because you have what I consider kind of elegant answer. Starts out kind of nasty looking, but turns out just one small little answer. I think it’s pretty neat.