Hi, and welcome. My name is Anthony Varela. And today, I'd like to talk about exponents and radicals in the order of operations. So we're going to review the order of operations, and then we're going to talk specifically about how to evaluate exponents and radicals. And then, we're going to practice evaluating expressions that contain exponents and radicals, following the proper order of operations.
So let's first review the order of operations. Many of us use PEMDAS to remember. And PEMDAS could be a mnemonic device, "Please Excuse My Dear Aunt Sally," to help remember those letters that make up PEMDAS. But what PEMDAS is is it's an acronym to help us remember the order of operations-- parentheses, exponents, multiplication, division, addition, and subtraction.
Now, recall that when we talk about parentheses in the order of operations, we're also including other grouping symbols. And when we're talking about exponents in the order of operations, we also call exponents "powers," but this also includes radicals or roots. And the big idea here, and the point of this lesson, is that we consider exponents and radicals at the same time, because any exponent can be written as a radical, and any radical can be written as an exponent. You can explore that in future lessons about radicals and exponents.
And then, of course, in the order of operations, we evaluate multiplication and division as we see them, reading left to right. And the same thing with addition and subtraction, as we see them, reading left to right. So the big idea for this lesson is that we are going to evaluate exponents and radicals before doing any other multiplication, division, addition, or subtraction outside of those terms.
So let's go ahead and look at how to evaluate expressions that have exponents using the correct order of operations. So here, I have an expression that reads 2 cubed plus the quantity 7 minus 4 squared. So the first thing I notice following the order of operations is that I have something in parentheses. That needs to be evaluated first.
So 7 minus 4 equals 3. So I'm going to rewrite this as 2 cubed plus 3 squared. So now I have a couple of exponents. And it doesn't matter in which order you evaluate these, as long as you evaluate them before doing anything else.
So I'm going to look at 2 cubed. And I know that that is 2 times 2 times 2, using that base 2 in a chain and multiplication three times. So 2 times 2 times 2 equals 8. So I'm going to write this as 8 plus 3 squared.
Still, I need to evaluate that other exponent before adding anything. So 3 squared means I'm taking 3 times 3. And 3 times 3 is 9. So now I have 8 plus 9, and I've cleared my exponents and can finish this up by adding. 8 and 9 gives us 17. So that's the value of my expression.
Now let's take a look at evaluating expressions with radicals using the correct order of operations. So here, my expression reads 18 minus 2 times the square root of 16. And my radical here is the square root of 16. So I don't see anything in parentheses, so I can jump right to that radical.
And then what that radical is telling me is that there needs to be a number that I can use two times in a chain of multiplication that will give me 16. And so I know that 4 times 4 is 16, so the square root of 16 is 4. So I'm going to go ahead and write in 4 for the square root of 16.
So now I have 18 minus 2 times 4. No more radicals, so now I can finish this up using the rest of my order of operations here. So what comes first, then? The subtraction or the multiplication? Well, that would be the multiplication.
So 2 times 4 is 8, so I'm going to write this as 18 minus 8. And now I can finish this up with subtraction, and this evaluates to 10. So let's go through a final example that combines all of these ideas here. So that looks pretty messy. Well, we're going to go through step by step, using our order of operations and evaluating some exponents and radicals and simplifying this.
So I have the quantity 5 minus 2 cubed plus 2 times the cubed root of 27. Now, I want to make one distinction here as I was writing this out. This does not read 2 cubed times 27. You should note that this 3 is part of the radical, so this is a cubed root here. I have 2 times the cubed root of 27. And we'll talk about the cubed root in a little bit.
So first, I see that there's something in parentheses that needs to be evaluated first. That's the quantity 5 minus 2. 5 minus 2 gives us 3. So I'm just going to rewrite this as 3 cubed plus 2 times the cubed root of 27.
And so now I'm going to evaluate my exponents and my radicals. And as I said before, it doesn't really matter in what order you do it-- they're not nested or anything-- just as long as we evaluate all exponents and all radicals before going on. So I like to just do it left to right, so we're going to take care of the 3 cubed, which is 3 times 3 times 3. And that equals 27. So I'm going to write this as 27 plus 2 times the cubed root of 27.
So now I don't have any more exponents, but I do have a radical. So I have to evaluate that before I go on. So my radical-- this is a cubed root, so I'm looking for a number that I can use three times in a chain of multiplication to arrive at 27. And I know that 3 times 3 times 3 equals 27.
And a little reminder-- that's what we just did over here. 3 times 3 times 3 got us to 27. So the cubed root of 27 is 3. So now I have 27 plus 2 times 3. And my multiplication comes next in the order of operations, so I'm going to evaluate 2 times 3. So now I have 27 plus 6, so I can complete this addition. And this evaluates to 33.
So that's how I used both exponents and radicals in a single expression to evaluate using the order of operations. So let's review our notes-- order of operations with exponents and radicals. Well, we talked about PEMDAS, which is an acronym to help remember the order of operations, which includes parentheses, exponents, multiplication and division, addition and subtraction.
We mentioned that the E for exponents includes radicals. And the reasoning behind that is because any exponent can be written as a radical, and any radical can be written as an exponent. So we consider both exponents and radicals in that step in PEMDAS, which means that we are going to evaluate all exponents and all radicals in our expressions before we compute any multiplication, division, addition, or subtraction. Well, thanks for watching this lesson on the order of operations with exponents and radicals. Hope to see you next time.