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## Algebra basics

### Course: Algebra basics>Unit 1

Lesson 5: Order of operations

# 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. At Sal 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?
• 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 at , you can not do:

1 + 2 - 3 + 4 - 1 = (1 + 2) - (3 + 4) - 1 = 3 - 7 - 1 = -5
• 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.
• 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.
• I have been taught BODMAS which is
Bracket
Of
Division
Multiplication
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😊
• 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.
• i'm a bit confused... :?
I live in england and my teacher told us to do:
Brackets (parentheses)
Indices (exponents)
Division
Multiplication
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????????
• PEMDAS = BIDMAS = BODMAS

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.
• Could you do order of operations with fractions?
• You can and should do it with everything from integers to decimals to fractions.
• is there an easier way to do it.
• No this is only way. Other methods are modified.
• 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
• 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!
• PEMDAS is what it would be for short
• So what Sal is explaining is that every single equation that you do will go in this order: Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction.
So let's say you have 3 x 8 + 18 / 2.
You have to solve it in the order of PEMDAS.
There are no parenthesis or exponents, so you have
MDAS.
First is multiplication, so you do 3x8 = 24.
Now you have 24 + 18 / 2.
18 / 2 = 9, so your equation is 24 + 9.
24 + 9 = 33
I really hope this helps you understand!
(1 vote)
• Please help me with math. Anything! I got a 67, and I need help. Specifically: Scientific Notation! And square roots. And pi.
• 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).