 This recording is going to talk about doing order of operations or what we call gem DOS. You might have heard it called PEMDOS, but I prefer gem DOS because the G used to be P for PEMDOS, which was parentheses, but really it could be any kind of grouping simple. So I like the G better. All right, so why do we have group order of operations? Well, if you look at this example two times three squared, there's several ways you could do it. You could think of that as two times three and then square it. Or you could think of it as two times three squared. If I take two times three and then square it, that's going to give me six squared, which is 36. So this says two times three squared, which is nine, and two times nine would be 18. Totally different answer. So we need an order of operations so that everybody does the same thing in the same order, and we'll all get the same answer. So let's look at a couple of examples. Remember, G means grouping symbols. So I look over here, and I have grouping symbols right here, but the only thing it has in it is a nine. So I don't have to worry about working inside the grouping symbols. E means exponent, and I don't have any exponents in this problem. And then M and D are both green. That means that we multiply and divide from left to right. It doesn't matter which one comes first. So I have a multiplication right here. In fact, I'm going to do it in green so that we see that that's what we're supposed to be doing here, is the multiplying and dividing. So we have three, plus, and then two times nine is 18. Now we've done all the multiplication and division, and we're ready to go and do the addition subtraction, and again, you do this from left to right. So we just have one addition here. So three plus 18 is going to be 21. So let's try the next problem. Again, we have to start with grouping symbols, and we don't have any. And then we look at exponents, and there's no exponents in this problem. And then we're going to do multiplication and division as we go across. So I have to do this multiplication before I can do that division or that multiplication right there. So eight times three is going to be 24, and then I will divide that by six, and then I actually will subtract, and then I will have two times two. Okay, so this is what we found. Now we're ready to do this part. So 24 divided by six is going to be four, and then I'm going to subtract, and I can do this multiplication. So I'm going to subtract the two times two, which is four, and then when I subtract four minus four, I get zero. If you can remember, you may want to put this at the top of your paper. You can just keep remembering that and work from left to right. You should be able to get these. Let's look at another set of examples. All right, we don't have the gemdoss up there, so let me get that up there. So grouping symbols, we have that right here. So let me just rewrite this problem. One-fifth times whatever we get there cubed, minus three-fifths is going to be equal to something. So in here, eight minus six is going to be two, and then we have the next thing that we would have had would be exponents, and we have an exponent right here. So if I recopy my problem, I have one-fifth, and then I'm going to do the two cubed, and two cubed would be, and this is still multiplying, though. One-fifth times this, two cubed happens to be eight, and then following the rest of the problem minus three-fifths. And then we had multiply and divide, and I have a multiplication right here. So one-fifth times eight, remember, we can write this as eight over one, so eight times one is eight, and five times one is five. So eight-fifths minus our three-fifths. So we're done with our multiplication and division, and then last we had addition and subtraction, and we have a subtraction right here. So eight-fifths minus three-fifths, or eight minus three over fifths, because they're all fifths. We have five over five or one. All right, they just keep getting more complicated, because when we do grouping symbols, let me just write it here. Grouping symbols means that we are going to do grouping symbols, that's what the G means, and then we have to do it from the inside out. So you want to do the innermost ones first. So if you look at this, we have brackets, and we have an absolute value, and we have parentheses. The parentheses are the innermost. Okay, so we have to do that part first. So two, and then 11, minus four, absolute value eight, minus three, and then just so that you can see that that's what we're doing right now. I'm going to do that in red, and then absolute value and bracket. So inside there five minus three is going to be two. We still have grouping symbols, so we have to do the G again. We had a grouping symbol here, and we still have a grouping symbol. So we would do our innermost first, so we're going to do this part right here, but inside there, this is a really complicated problem. Inside there, we have to look at our multiplication. All right, so let's see what we can do here. Two, bracket, 11, minus four, absolute value, which is a grouping symbol, and then eight, minus, and I have to do my multiplication. Okay, in fact, I'm going to do that in green, since we're used to multiplication being green, and three times two is six, and then our absolute value and our bracket. All right, so now we have to, again, we have a grouping symbol here that we have to deal with, right here. So if we continue writing the problem, two times eleven, or times a bracket, eleven minus four, absolute value, and then we're going to fill that in, and when we fill that in, it's an addition or subtraction, so eight minus six is two. Okay, we subtracted, so we got that, and now we have another grouping symbol, because we've taken care of the absolute value, but we still have this bracket. Okay, so this grouping symbol at the beginning was a parentheses, and then we had the absolute value, but we had to keep working on the absolute value, and now we're going to do the bracket. So we have writing our problem, two, bracket, eleven minus four, absolute value of two, bracket. Just to make it look simpler, we should do this part right here first, and remember that the absolute value of two, I'm just going to come in here, when we take it, the absolute value of two, what is the absolute value of two? The absolute value of two is two. So it's really times two, and then that's a multiplication that we need to do next, because inside the parentheses, or the brackets, or whatever your grouping symbols are, you have to follow the gemdoss again, until you get that simplified. So we have two bracket, eleven, and then minus, and four times two is going to be eight, so we have, we're still dealing with our grouping symbol, and now we can do the subtraction inside there. So we have rewriting the problem, two, and then inside this bracket, we're going to do our subtraction, eleven minus eight is three, and then we have our multiplication, which is two times three equals six. One last problem. Wow, look at this. So for our grouping symbols, we have a square root, and we have a division bar, which equals division. So those are the first things that we have to do. So we're going to take care of simplifying, we have to do all of this first, and then we'll go from there. So ten, minus, and then we have the square root, and you'll notice there's an exponent in here, and there's multiplication in here, so we've got a couple things going on here. We've got, I'm going to write it in colors so that we know what we're going to do. So ten squared is our exponent. Minus will be our subtraction, which should have been purple. Minus is going to be subtraction, and then we have four, times two, times eight, and then we've got our grouping symbol. Actually, this was a grouping symbol, and then we've got our two times eight. So let's go and see what we can do. Ten minus, and then we still have a square root, and if we do our ten squared, that's one hundred, and then subtract, and then I'm going to do these two at a time. So four times two is eight, times eight, all over, and you're kind of doing several things at a time because we did it in colors. And you might want to just do it in dark pencil and light pencil or something to that effect, or brings in colored pencils. I'm just going to leave this two times eight down here. We're just dealing with the grouping symbol right now. So continuing on, ten minus the square root of, we've got our one hundred, and we're going to subtract the eight times eight, which is sixty four, and then that's a grouping symbol. This up here, if I wanted to really follow my colors, the division sign is a grouping symbol, and then I've got two times eight. So continuing on, ten minus the square root of, a hundred minus sixty four, and that happens to be thirty-six, and then divided by two times eight. So if we follow along, we're going to need some more space here. We've got ten minus the square root of thirty-six now is six, and then divided by and two times eight. We'll do that one next. Okay, so now we're ready to do our multiplication and division, because we've done the grouping symbols, we've done the exponents. This is a multiplication. The division here, actually we have to take care of the subtraction on the top. We have to take care of the multiplication on the bottom, and then we can actually do the division, because we need it to be simplified. This means simplify the top, grouping symbol, simplify the bottom. So we have ten minus six, which is going to be four, and two times eight is going to be sixteen, and then we finally get to divide that, and four divided by sixteen is actually one over four, and there you have it.