Today, we're going to talk about grouping symbols, which are a type of parentheses in our order of operations. So we'll start by reviewing PEMDAS, and then we'll do some examples with the different types of grouping symbols.
So let's start with the review of PEMDAS, which is an acronym that we use to remember the order of operations. And PEMDAS stands for Parentheses, Exponents, Multiplying, Dividing, and Adding and Subtracting. So our topic for today is actually focusing on parentheses, which includes lots of different types of grouping symbols.
So we have our standard parentheses, but we also have absolute value signs, fraction bars-- so you could have an expression in the numerator and in the denominator-- and also, square root symbols. So all of these are considered grouping symbols, and should be evaluated with the P for Parentheses. In other words, evaluated first.
Then, we've got exponents, which include things like this, something squared. But exponents also include radicals, like square roots and cubed roots.
Then, we've got multiplying and dividing, adding and subtracting. And the thing that we want to always remembers for multiplying and dividing is that they should be done together from left to right. And similarly for adding and subtracting, we want to remember that they should also be done together, from left to right.
So now that we've reviewed PEMDAS, let's do some examples using our different types of grouping symbols.
So here's my first example. I've got an absolute value sign, which we know acts as a grouping symbol. So we're going to evaluate that with the P under Parentheses, meaning we're going to start with that first. And inside my absolute value sign, I've got adding and multiplying.
Thinking about PEMDAS, I know we need to multiply first. So 3 times negative 5 is going to give me a negative 15, and I'm going to add to that the 12 inside my absolute value sign.
I'm not going to bring down the rest of the problem until I finish evaluating what's inside the absolute value. So 12 plus negative 15 is going to give me a negative 3, still inside my absolute value sign. And the absolute value of negative 3 is just going to give me 3.
So now that I've evaluated this, I'm going to bring down the rest of my problem. I've got 2 times 9 plus-- and then behind it, minus 6 divided by 2.
So we've got multiplying, adding, subtracting, and dividing. Thinking of order of operations, I know I need to multiply and divide from left to right. So 2 times 9 is going to give me 18. Bring down the 3. And 6 divided by 2 is also going to give me 3.
So now, I have adding and subtracting left. And I'm going to evaluate from left to right. So 18 plus 3 is going to give me 21, and subtract 3 is going to bring me back to 18.
So here's my second example. Here, I've got a fraction bar, which we know is a type of grouping symbol. So that means we need to start by evaluating it first. And the way we do that is by evaluating the numerator together, and then evaluating the denominator. And then, we'll perform the operation of the fraction bar, which is to divide.
So starting with my numerator, I've got 18 minus 4 times negative 3. Between subtracting and multiplying, I know I need to multiply first. So bringing this down, I've got 18 minus-- 4 times negative 3 is going to give me a negative 12. And I'm just going to rewrite my denominator, and add the plus 6 at the end.
So now, I need to finish evaluating my numerator. 18 minus a negative 12 is going to give me a 30. And underneath, I still have my 2 times negative 5 and my plus 6 at the end.
So now that I've evaluated my numerator, I can evaluate my denominator. 2 times negative 5 is going to give me a negative 10. So now, I have 30 over negative 10, with my plus 6 at the end.
So now that I've evaluated my fraction bar, I can divide. 30 divided by negative 10 is going to give me a negative 3, and I'm adding my 6 to that. That was at the end. Negative 3 plus 6 is going to give me a positive 3 for my final answer.
So here's my third example. I've got a radical sign, which I know acts as a grouping symbol, which means that I need to evaluate what's underneath the radical first. If you're feeling confident with using the order of operations, go ahead and pause and try this on your own. And then, check back and see how you did.
So starting underneath my radical, I've got multiplying and adding. I know that I need to multiply first. So 4 times 2 is going to give me 8, and bring down my plus 1. I'll bring down the rest of this problem later.
Underneath my radical, 8 plus 1 is going to give me 9. So now, I have the square root of 9. And the square root of 9 is just 3. So now that I've evaluated my radical, I'm going to bring down the rest of my problem. So negative 6 divided by 2 plus-- and then, my 3.
So now I've got dividing and adding. I know that I need to divide before I add. So negative 6 divided by 2 is going to give me a negative 3. And when I add my 3 to it, that's going to give me a final answer of 0.
So for my fourth example, I've got two sets of grouping symbols, my parentheses and my absolute value sign. I'm going to start by the innermost grouping symbol, which is my parentheses.
So 5 minus 1 is going to give me 4. Bring down the rest of my problem, and I'll leave this for later. Continuing inside my absolute value, 100 divided by 4 is 25, and the absolute value of 25 is just 25. But we still need to bring down this negative that was in front of it.
So now that I've evaluated my absolute value sign, I'm going to bring down the rest of my problem. So I know that I need to evaluate my radical first. So the square root of 64 is going to give me 8. That's going to be multiplied by the 5 that's in front.
So now, I have subtracting and multiplying. I know I need to multiply first. 5 times 8 is going to give me 40. And now that I only have subtracting, negative 25 minus 40 is going to give me negative 65.
So let's go over our key notes from today. We talked about grouping symbols, and the fact that they fall under parentheses, which means that when you're evaluating with the order of operations, you're going to start with grouping symbols first. And those grouping symbols include things like an absolute value sign, a fraction bar, or a radical.
And we also talked about nested grouping symbols, where you have more than one grouping symbol inside of another one. And we saw that you need to start by evaluating them from the inside out. So whatever your innermost grouping symbol is, start with that, and then work your way out.
So I hope that these examples and these notes helped you understand a little bit more about grouping symbols and the order of operations. Keep practicing. Keep using your notes, and soon, you'll be a pro. Thanks for watching.
An acronym to remember the order of operations: parentheses, exponents, multiplication and division, addition and subtraction.