Welcome to order of operations. The goal of this video will be to simplify expressions using
the order of operations.
Now the order of operations are probably one of the most important
set of rules you need to follow when performing mathematical calculations.
It's kind of like the rules of the game. If everyone doesn't calculate
mathematical procedures in the same way then we would not
obtain the same answers, so for consistency it's extremely important
that we follow the rules, or the order of operations.
And here they are. Number one: you to calculate what's in the innermost
parenthesis, or grouping symbols.
And these grouping symbols can take on a variety of forms.
Number two: we need to simplify all expressions with exponents.
Numbers three: multiply and divide from left to right.
And then number four: add and subtract from left to right.
Now there are a lot of acronyms, or ways to remember the order of operations.
I'm not a big fan of them, but here's a couple of them. One acronym is called
PEMDAS. P-E-M-D-A-S.
And you can see P for parenthesis, E for exponents,
M for multiplication and so on. You've probably also heard of please excuse my dear Aunt Sally,
just a phrase to help you remember the acronym PEMDAS.
Now the reason I don't like this is that it does look like there are six steps to
the order of operations,
but in fact there are only four. We have parenthesis,
exponents, multiplication and division have the same priority,
so you perform multiplication and division from left to right. You do not multiply first
and then divide.
And the same thing, the last two have the same priority,
so there's really only four steps. So it's okay to use these shortcuts to remember
these steps,
but just remember there are only four, not six steps.
Well the best thing to do is go ahead and give these a try. Simplify the
following expression.
it's very tempting to work this from left to right, but first off there's no
parenthesis,
second there's no exponents, so step three is multiply and divide.
So we have to perform this multiplication here
before we add and subtract. So if we do that we'd have twelve
minus two times four of course is eight, plus nine.
Now subtraction and addition have the same priorities so we work from left to right.
The twelve minus eight would give us four, plus the nine
is equal to thirteen. Let's take a look at a few more.
Here you can see it gets more involved rather quickly,
and step one is work within the innermost parenthesis,
so we need to perform this operation first,
And it's very helpful to rewrite this each time instead of trying to make shortcuts.
So we have seven minus nine which would give us a negative
two, third power. Now we're still working within the innermost parenthesis,
so since we have an exponent here we have to cube negative two.
Of course negative two to the third power would be negative eight.
Still working within the parenthesis, now we have addition and multiplication.
Multiplication takes priority over addition, so the next step, still inside the parenthesis,
three times negative eight would be a negative twenty-four.
Again still working within the parenthesis,
we have to simplify fifteen plus negative twenty-four.
Fifteen plus negative twenty-four would give us
negative nine. Next step we have division addition and multiplication.
Multiplication and division take priority over addition so we multiply and divide from
left to right.
So six divided by two, that would give us three,
plus three times negative nine would give us negative twenty-seven,
and then finally we end up with negative twenty-four.
So we're showing a lot of work here, but it's definitely going to be worth it to
make sure we get the correct answer.
Next one, we have a lot going on.
Again we start with the innermost parenthesis, so we're going to start
here. So negative three plus seven would give us positive four.
Still working within the innermost parenthesis, we need to simplify this part.
Of course four minus one would give us three.
Next step we need to simplify our
exponents here,
and here. Four squared is sixteen,
two cubed is eight. Our next step is to perform
multiplication and division as it occurs from left to right.
So here we have some division, here we have multiplication.
Twenty divided by five would give us four,
sixteen times three would give us forty-eight.
And the last step, add and substract as it occurs from left to right.
Four plus forty-eight would be fifty-two,
minus eight should give us forty-four.
I think I have one more. When you have an expression in fraction form,
we need to simplify the numerator and denominator separately before we deal with the fraction,
that is mentioned in step one of the order of operations.
So let's perform the operations in the parenthesis in the numerator.
Six plus four of course would be ten. Six plus four is ten.
Seven minus five would be two.
In the denominator we have for minus two, of course that's two.
Step two in the order of operations would be exponents.
So we'll simplify this in the numerator and this in the denominator.
Two to the fifth power is equal to thirty-two.
Two to the second power is equal to four.
So we're treating the numerator and the denominator as if they were their own problems,
and then we'll simplify the fraction at the end.
So step three the order of operations, multiply and divide as it occurs from left to right.
So here we have multiplication and here we have multiplication as well.
So we have forty plus thirty-two and twelve minus four.
So it looks like we have seventy-two divided by eight,
which of course is equal to nine. I hope this overview of the order of operations
helps you to simplify expressions properly.