Hi, and welcome back. My name is Anthony Varela, and today in this lesson, we're going to talk about operations that act as grouping symbols. So we're going to review the order of operations, and then we're going to talk about some operations that imply parentheses. And that's going to affect how we evaluate our problems. And then we're going to practice evaluating these expressions and where these operations that imply parentheses pop up.
So first, let's review the order of operations. Now, a common acronym is PEMDAS. And a common mnemonic device is "Please Excuse My Dear Aunt Sally" to remember these letters that make up this acronym. And this acronym is a helpful acronym to remember the order of operations, which are Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction.
Now, a couple of notes about these operations is that when we're talking about parentheses in the order of operations, this also includes other grouping symbols. And we're going to talk about those today. Exponents also includes radicals. And exponents could also be called powers, and radicals could also be called roots.
Now, with multiplication and division, we perform these as we see them, reading left to right, so not always multiplication before division. And the same is true with addition and subtraction. We perform these as we see them, reading left to right, not always addition before subtraction.
So now let's talk about some operations that act as grouping symbols. And specifically, we're going to be talking about absolute value bars, fraction bars, and radicals. So we're going to take a look at each three of these individually.
So absolute value bars as grouping symbols-- so here's an example of an expression that has absolute value bars. And notice that we have an expression inside of those absolute value bars. We have 6 minus 10. So we can think of this as being a set of parentheses. And we have to evaluate that first, even though I see multiplication, which I know becomes comes before addition and subtraction in the order of operations. Here, I have implied parentheses.
So to evaluate this, first, we take 6 minus 10. That is negative 4. So we have 2 plus 8 times the absolute value of negative 4. Well, we know that the absolute value of negative 4 is positive 4. So we have 2 plus 8 times 4.
And now we're going to perform the multiplication before our addition. So we have 2 plus 32. So this evaluates to 34. So our entire expression evaluated to 34.
Now, let's take a look at the fraction bars as grouping symbols. So here's what that looks like. Here, I have an expression in my numerator and an expression in my denominator. So one way to think of this is that there are implied parentheses that group our entire numerator together, and a set of parentheses that groups our entire denominator together. So we have to evaluate these separately before we actually do the division that our fraction bar represents. So this division by the fraction bar actually comes last in our order of operations.
So let's go ahead and evaluate, then, the numerator. So we know that 4 times 5 is 20. And then we're taking away 8. Remember, we're doing the 4 times 5 first, because multiplication comes before subtraction in our order of operations. So our numerator is 20 minus 8. And that equals 12.
So now let's put our denominator into all of this. So we have 16 minus 6 divided by 3. What are we going to do first, that division or the subtraction?
Well, we're going to do the division first. So 6 divided by 3 equals 2. So I know that my denominator is 16 minus 2. So my denominator, that is 14. So my whole expression has now been simplified to 12/14, which can be simplifying even further to 6/7.
Now let's take a look at radicals that can act as grouping symbols. So here's an example of a radical that implies parentheses. Now, notice we have an expression that's underneath the radical. Notice that we have something outside of the radical, too. So this does not get put in the implied parentheses. This is what our implied parentheses look like.
So we actually have to evaluate 1 plus 6 times 4 first. And then we'll return to our order of operations, because this is all part of grouping symbols, which happens first.
So taking a look at 1 plus 6 times 4, we do the multiplication first. So we have 1 plus 24 underneath our radical, and then we are subtracting a 7 outside of that. So 1 plus 24 is 25. So we have the square root of 25 minus 7. Well, 25, luckily for us, is a perfect square, so the square root of 25 equals 5. So now we're simply just evaluating 5 minus 7, which is negative 2.
All right, well, let's take a look at a final example that's really going to combine a lot of things. So we have this expression that looks quite messy, to be honest. But we're going to apply what we know about the order of operations, what we know about operations that imply parentheses, so these absolute value bars, these fraction bars, these radicals. And we're really going to take this step by step.
Well, I notice that we have something new in here as well, which we'll get to. We have a nested operations here. We have a set in parentheses, and then we have some brackets, too. So we're going to take a look at that as we go through this. I also notice that we have fraction bars, and we have a radical.
Now, I'm going to focus on the fraction bar here, because what this really does is it separates an entire numerator which has lots of junk, and an entire denominator which has lots of junk. So we're actually going to evaluate these separately. I have made a nice little line one side. We're going to work through our entire numerator. And then on the other side, we're going to work through our entire denominator.
So let's work with the numerator. Well, we have 18 minus 5. That's set inside parentheses. Then we're going to multiply that by 2. All of that is within these brackets. And then we have a minus 5 outside of everything.
So I'm introducing here nested operations. And basically, the idea here is that we're going to work from the inside out. So we're going to evaluate the 18 minus 5. And then we're going to apply the multiplying by 2. And then we'll take care of everything else.
So 18 minus 5, we could rewrite as 13. So we have 13 times 2. That's still within these brackets, which is a grouping symbol-- needs to be evaluated before we do anything else. So 13 times 2 is 26, and then we'll take away 5. So 26 minus five is 21, so that is our entire numerator. All of this up here in our numerator just equals 21.
Now let's work with our denominator. So our denominator is the square root of 18 divided by 2. And then we have a plus 4 outside of that radical sign. So here, this radical sign has implied parentheses. We need to evaluate this 18 minus 2 first-- or 18 divided by 2, sorry.
So 18 divided by 2 equals 9. So we have the square root of 9 plus 4. Now, the square root of 9 equals 3. And then we're going to add 4 to that. So our entire denominator, then, simplifies to 7.
So I'm going to then return to my original example. And I know that the numerator is 21. The denominator is 7. So we have 21 divided by 7, doing that fraction bar last. And 21 divided by 7 equals 3. So that's how we can use all of this information about the order of operations and different grouping symbols that imply parentheses to solve a pretty complicated problem.
So let's review our notes. We talked about the order of operations, and we talked about PEMDAS, which is an acronym to remember the order of operations-- Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. But we talked about operations that can act as grouping symbols or implied parentheses, specifically absolute value bars, fraction bars, and radicals.
We also talked about nested operations. For example, here was where we had 18 minus 5 within another grouping symbol of a quantity multiplied by 2. And the whole idea here was to work from the inside out. So if you remember first, we evaluated the 18 minus 5. Then we multiplied it by 2. So thanks for watching this video about operations as grouping symbols. Hope to catch you next time.