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### Course: 7th grade (Illustrative Mathematics) > Unit 5

Lesson 14: Lesson 14: Solving problems with rational numbers# Intro to order of operations

This example shows the steps and clarifies the purpose of order of operations: to have ONE way to interpret a mathematical statement. Created by Sal Khan.

## Want to join the conversation?

- I have always thought that within the same level of priority that the specific order (left to right, right to left, jumping around, etc.) wasn't important. At5:40Sal says that you have to do things from left to right when you have multiple operations at the same level. At this point in the video, the problem is: 10 x 4 / 2 - 5 x 6

Sal solves left to right: 40 / 2 - 5 x 6 = 20 - 30 = -10

But if I don't do it in the same order I get the same answer: 10 x 2 - 5 x 6 = 20 - 30 = -10

Thoughts?(180 votes)- This confused me when Sal first said it too, but it can make a difference. For example, if the question were rearranged to:

10 / 2 x 4 - 5 x 6

Then you can't do 2 x 4 first i.e:

10 / (2 x 4) - 5 x 6

Otherwise you would get:

10 / 8 - 5 x 6

1.25 - 30

-28.75

Similarly, in the example at7:50, you can**not**do:

1 + 2 - 3 + 4 - 1 = (1 + 2) - (3 + 4) - 1 = 3 - 7 - 1 = -5(178 votes)

- The practice questions expect you to accept that a fraction bar is the equivalent of putting parentheses around the whole numerator and the whole denominator. Did Sal cover this in either of the order of ops vids? I can't find it but maybe I missed it. If not, would be a good addition to the vids.(82 votes)
- I'm not sure that he covered this in the video, but when you have multiple operations over a fraction bar, with more operations or a single number underneath, the implication is that you are dividing the entire operation by the number underneath the fraction bar (fractions are essentially saying "the numerator divided by the denominator"). You cannot divide the operation until you have solved it, of course, so it is implied in the layout of the equation itself that you need to solve the numerator and/or denominator before dividing.(22 votes)

- I have been taught BODMAS which is

Bracket

Of

Division

Multiplication

Addition

Subtraction .

This is mostly the same as brackets and parentheses are the same and exponents is a different thing but then am I supposed to do multiplication first or division ??

I have been taught that I have to divide first but here they have explained something else . What do I have to do ?

All help appreciated😊(24 votes)- The way I have been taught is with PEMDAS; parenthesis, exponent, multiplication, division, addition, and subtraction. When it comes to multiplication and division, you do whichever comes first in a left to right order, same goes for addition and subtraction.(27 votes)

- i'm a bit confused... :?

I live in england and my teacher told us to do:

Brackets (parentheses)

Indices (exponents)

Division

Multiplication

Addition

Subtraction

...so i dont do add and sub in the same group and if they are together go from left to right coz i would do the addition then the subtraction... is it different over in the US???? plz i am going mad thinking about it, which one is right????????(17 votes)- PEMDAS = BIDMAS = BODMAS

PEMDAS*parentheses; exponents; mult/div; add/sub*

BIDMAS*brackets (parentheses); indices (exponents); div/mult (mult/div); add/sub*

BEDMAS*brackets (parentheses); exponents; div/mult (mult/div); add/sub*

They're all the same way of order of operations. It's just that people use other words to tell the same thing.(7 votes)

- This is BEDMAS:-

B = Brackets = Rank 1

E = Exponents = Rank 2

D = Division = Rank 3

M = Multiplication = Rank 3

A = Addition = Rank 4

S = Subtraction = Rank 4

BEDMAS - That's how I remember it

Always go left to right when doing the same rank.

There is also GEMS:-

G = Groupings = Rank 1

E = Exponents = Rank 2

M = Multiply/Divide = Rank 3

S = Subtract/Add = Rank 4

This is basically the same thing as Bedmas!!

Please upvote;)(14 votes) - PEMDAS is what it would be for short(10 votes)
- Please help me with math. Anything! I got a 67, and I need help. Specifically: Scientific Notation! And square roots. And pi.(4 votes)
**Square roots**:

Square roots are basically the “inverse operation of 2nd powers”. Remember that when something is raised to the second power, it is that value of the base multiplied by itself once. For example, 7^2 equals 7x7, which equals 49. In other words, whatever x-value raised to the 2nd power, if you apply the square root to it the result will return to the original x-value. The square root of 49 will equal 7. The square root of 4 will equal 2. When you have the square root function, it basically means it is wanting you to go back to the value which, when multiplied by itself, will reach the value inside the square root.

The expression “√x=y” is telling you, x = y^2, and√64=8, because you are being asked, “What value squared equals 64?” The answer is 8, because 8^2=64. For a constant, n, or when rewritten into the expression x = y^2, it is “What value do you need to plug into the y (your answer) so you get the amount displayed inside the square root?”

• Square roots for most values, except those which already are perfect squares, will result in an irrational value.

• If you haven't already, practice and learn multiplication and division, as well as prime factorization. Next, memorize the squares of all the single-digit numbers (I would recommend that you do so such that you eventually am able to instantly recall the square of any single-digit in less than a second). They will greatly help in memorizing which values are perfect squares. You can also try memorizing the squares of a few 2-digit numbers. Then, practice simplifying square roots through prime factorization. Even if your calculator is not designed for engineering and can only give decimal approximations, as long as there is the square root function you should be fine. Do practices of one value subtracted by a simplified version and see if the difference is 0 or something very close to it (E.g. √8 minus 2√2, √125 minus 5√5, etc.).

• After you are proficient (or even a "master") at recognizing perfect squares and their square roots, first memorizing some decimal approximations of simple values, such as √2, and use those values to help make estimates of non-perfect-squares.

• Here are some values squared (the arrow is pointing towards the result after the original value is applied the second power; if you want, I can add more values and their squares into this list. Just post a comment to let me know).

•1=>1

•2=>4

•3=>9

•4=>16

•5=>25

•6=>36

•7=>49

•8=>64

•9=>81

•10=>100

•11=>121

•12=>144

•13=>169

•14=>196

•15=>225

•16=>256

•17=>289

•18=>324

•19=>361

•20=>400

•21=>441

•22=>484

•23=>529

•24=>576

•25=>625

•26=>676

•27=>729

•28=>784

•29=>841

•30=>900

•31=>961

•32=>1024

•33=>1089

•34=>1056

•35=>1225

•36=>1296

•37=>1369

•38=>1444

•39=>1521

•40=>1600

Therefore, things are working the other way around: If you have say √1024, then your answer will be 32 because 32, when squared, gives you 1024. By memorizing a number of perfect squares, you can check if the value within the square root is a perfect square, and if it is not find the two perfect squares nearest to it, one greater and one less. For example, if you have √1500 [a decimal approximation is 38.7298, you may be able to quickly see that since 1500 is between 1444 and 1521, then the square root of 1500 is between 38 and 39.

(This is the same way cube roots and other radicals/roots work, except that cube roots use 3rd powers, etc.)**Simplifying square roots by factoring**: Start by factoring perfect squares out of the value displayed within the square root symbol. For example, if you have √50, you can factor it into √(25x2). Whatever values are perfect squares may be moved out of the square root sign while turning that perfect square into the square root (since the value is “no longer being affected by the square root”. √(25x2) equals 5 x √(2), approximately 7.07. However, remember that the square root values outside are multiplied together (and not added). Finally, whatever factors which are not perfect squares are left within the square root sign. If there is more than one factor, then those get multiplied within the square root. For example, if there is √1800, you can do it in these steps (the xs are multiplication signs, not variables):

•Factor a 100 (or two 10s) out of the 1800, so you have √(100 x 18)

•Factor a 9 (or two 3s) out of the 18, so you have √(100 x 3 x 3 x 2)

•Applying the square root to 100 means you have 10, and applying the square root to 9 means you have 3. (Remember that for square roots, whatever “pair of two equal numbers” means a perfect square.)

•10 x 3 = 30; since there is only a single 2, not a pair, you end up with 30 x √(2).(11 votes)

- Could you do order of operations with fractions?(7 votes)
- You can and should do it with everything from integers to decimals to fractions.(4 votes)

- Uhm hi uhh I wanna know, uhh what is the meaning of "Exponents" It is a hard word to remember and spell. Can't they just eliminate it?

Thanks, Sal

- Lexi! <3(5 votes)- Hi Lexi! I am not Sal, but I can still help you understand exponents if you want to.

Exponents are numbers like this: 10⁵

The big number is the base while the smaller number floating is the number of times you multiply a base by itself. For example, in the exponent "10⁵", the expanded sentence is 10*10*10*10*10, which is 100,000. A trick only in exponents when 10 is the base is the number of 0's in the value is the small number. Like in the example, 10⁵, 5 is the small number or the exponent. So there will be 5 0's in the answer, 100,000. in exponents when 2 is the base, you just double the number the amount of times the small number, or the exponent is. For example, if the problem is 2⁵, I double 2, 4 times to get 32 (The first time doesn't count because 2¹ is just 2). also if 1 is the base, no matter what the small number is, the answer is always 1. and if the small number is 1, then the answer or value is always the base.

Hope this helps!(10 votes)

- is there an easier way to do it.(5 votes)
- No this is only way. Other methods are modified.(9 votes)

## Video transcript

In this video we're going
to talk a little bit about order of operations. And I want you to pay close
attention because really everything else that you're
going to do in mathematics is going to be based on you
having a solid grounding in order of operations. So what do we even mean when
we say order of operations? So let me give you an example. The whole point is so that we
have one way to interpret a mathematical statement. So let's say I have the
mathematical statement 7 plus 3 times 5. Now if we didn't all agree on
order of operations, there would be two ways of
interpreting this statement. You could just read it left to
right, so you could say well, let me just take 7 plus 3, you
could say 7 plus 3 and then multiply that times 5. And 7 plus 3 is 10, and then
you multiply that by 5. 10 times 5, it
would get you 50. So that's one way you would
interpret it if we didn't agree on an order of operations. Maybe it's a natural way. You just go left to right. Another way you could interpret
it you say, I like to do multiplication before
I do addition. So you might interpret it as --
I'll try to color code it -- 7 plus -- and you do
the 3 times 5 first. 7 plus 3 times 5, which would
be 7 plus 3 times 5 is 15, and 7 plus 15 is 22. So notice, we interpreted
this statement in two different ways. This was just straight left
to right doing addition then the multiplication. This way we did the
multiplication first then the addition, we got two different
answers, and that's just not cool in mathematics. If this was part of some effort
to send something to the moon because two people interpreted
it a different way or another one computer interpreted one
way and another computer interpreted it another way, the
satellite might go to mars. So this is just completely
unacceptable, and that's why we have to have an agreed
upon order of operations. An agreed upon way to
interpret this statement. So the agreed upon order of
operations is to do parentheses first -- let me write it over
here -- then do exponents. If you don't know what
exponents are don't worry about it right now. In this video we're not going
to have any exponents in our examples, so you don't
really have to worry about them for this video. Then you do multiplication --
I'll just right mult, short for multiplication -- then you do
multiplication and division next, they kind of have the
same level of priority. And then finally you do
addition and subtraction. So what does this order of
operations -- let me label it -- this right here,
that is the agreed upon order of operations. If we follow these order of
operations we should always get to the same answer
for a given statement. So what does this tell us? What is the best way to
interpret this up here? Well we have no parentheses --
parentheses look like that. Those little curly
things around numbers. We don't have any
parentheses here. I'll do some examples that
do have parentheses. We don't have any
exponents here. But we do have some
multiplication and division or we actually just have
some multiplication. So we'll order of operations,
do the multiplication and division first. So it says do the
multiplication first. That's a multiplication. So it says do this
operation first. It gets priority over
addition or subtraction. So if we do this first we
get the 3 times 5, which is 15, and then we add the 7. The addition or subtraction --
I'll do it here, addition, we just have addition. Just like that. So we do the multiplication
first, get 15, then add the 7, 22. So based upon the agreed order
of operations, this right here is the correct answer. The correct way to
interpret this statement. Let's do another example. I think it'll make things a
little bit more clear, and I'll do the example in pink. So let's say I have 7 plus 3 --
I'll put some parentheses there -- times 4 divided by
2 minus 5 times 6. So there's all sorts of crazy
things here, but if you just follow the order of operations
you'll simplify it in a very clean way and hopefully we'll
all get the same answer. So let's just follow the
order of operations. The first thing we have to
do is look for parentheses. Are there parentheses here? Yes, there are. There's parentheses
around the 7 plus 3. So it says let's do that first. So 7 plus 3 is 10. So this we can simplify,
just looking at this order operations, to
10 times all of that. Let me copy and paste
that so I don't have to keep re-writing it. So that simplifies to
10 times all of that. We did our parentheses first. Then what do we do? There are no more parentheses
in this expression. Then we should do exponents. I don't see any exponents here,
and if you're curious what exponents look like, an
exponent would look like 7 squared. You'd see these little small
numbers up in the top right. We don't have any exponents
here so we don't have to worry about it. Then it says to do
multiplication and division next. So where do we see
multiplication? We have a multiplication,
a division, a multiplication again. Now, when you have multiple
operations at the same level, when our order of operations,
multiplication and division are the same level, then
you do left to right. So in this situation you're
going to multiply by 4 and then divide by 2. You won't multiply
by 4 divided by 2. Then we'll do the 5 times
6 before we do the subtraction right here. So let's figure
out what this is. So we'll do this
multiplication first. We could simultaneously do this
multiplication because it's not going to change things. But I'll do things
one step at a time. So the next step we're going
to do is this 10 times 4. 10 times 4 is 40. 10 times 4 is 40, then you
have 40 divided by 2 and it simplifies to that right there. Remember, multiplication and
division, they're at the exact same level so we're going
to do it left to right. You could also express this as
multiplying by 1/2 and then it wouldn't matter the order. But for simplicity,
multiplication and division go left to right. So then you have 40 divided
by 2 minus 5 times 6. So, division, you just
have one division here, you want to do that. You have this division and you
have this multiplication, they're not together so you
can actually kind of do them simultaneously. And to make it clear that you
do this before you do the subtraction because
multiplication and division take priority over addition and
subtraction, we could put parentheses around them to say
look, we're going to do that and that first before I do that
subtraction, because multiplication and
division have priority. So 40 divided by 2 is 20. We're going to have that minus
sign, minus 5 times 6 is 30. 20 minus 30 is equal
to negative 10. And that is the correct
interpretation of that. So I want to make something
very, very, very clear. If you have things at the same
level, so if you have 1 plus 2 minus 3 plus 4 minus 1. So addition and subtraction are
all the same level in order of operations, you should
go left to right. So you should interpret this as
1 plus 2 is 3, so this is the same thing as 3 minus
3 plus 4 minus 1. Then you do 3 minus 3
is 0 plus 4 minus 1. Or this is the same thing
as 4 minus 1, which is the same thing as 3. You just go left to right. Same thing if you have
multiplication and division, they're at the same level. So if you have 4 times 2
divided by 3 times 2, you do 4 times 2 is 8
divided by 3 times 2. And you say 8 divided by 3 is,
well, we got a fraction there. It would be 8/3. So this would be 8/3 times 2. And then 8/3 times to
is equal to 16 over 3. That's how you interpret it. You don't do this
multiplication first or divide the 2 by that and all of that. Now the one time where you can
be loosey-goosey with order of operations, if you have all
addition or all multiplication. So if you have 1 plus 5 plus 7
plus 3 plus 2, it does not matter what order you do it in. You can do the 2 plus 3, you
can go from the right to the left, you can go from the
left to the right, you could start some place in between. If it's only all addition. And the same thing is true if
you have all multiplication. It's 1 times 5 times
7 times 3 times 2. It does not matter what
order you're doing it. But it's only with all
multiplication or all addition. If there was some division
in here, if there's some subtraction in here, you're
best off just going left to right.