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Evaluating expressions with & without parentheses

Explore the concept of order of operations, emphasizing the importance of parentheses in mathematical expressions.Learn how the placement of parentheses can drastically change the outcome of an expression, highlighting the need to follow the correct sequence: parentheses, multiplication and division, then addition and subtraction. Created by Sal Khan.

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  • primosaur ultimate style avatar for user Grace_Chen
    What's the difference between parentheses(), curly brackets{}, and brackets[]?
    (3 votes)
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    • purple pi purple style avatar for user signa1
      Parentheses () are used to group numbers or expressions that you want to calculate first. For example, in 2 + (3 × 4), you do 3 × 4 first because it is inside the parentheses.
      Curly brackets {} are used to show sets of numbers or objects. A set is a collection of things that have something in common. For example, {1, 2, 3} is a set of three numbers.
      Brackets [] are basically only used when you have more than one pair of parentheses in an expression. You use them outside the parentheses to show what to do next. For example, in [2 + (3 × 4)] ÷ 5 , you do 2 + (3 × 4) first because it is inside the brackets.

      In a complicated expression, all three can be used, in this order: parentheses first, then brackets, then curly brackets. For example:

      4 - 3 [4 - 2 {6 - (5 + 1)}] ÷ 3
      First: 4 -3 [4 -2 {6 - 6}] ÷ 3
      Then: 4 -3 [4 -2 × 0] ÷ 3
      Then: 4 - 3 × 4 ÷ 3
      Then: -8 ÷ 3
      So the answer is -8 divided by 3.
      (11 votes)
  • starky sapling style avatar for user Dagoberto
    why does the math seem easier in the videos than in reality
    (78 votes)
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  • starky sapling style avatar for user 26noelbrooklyn
    can you please show more examples because those two were helpful and i did understand it but it is still pretty confusing
    (20 votes)
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    • starky ultimate style avatar for user Anonymous Phoenix
      Hi 26noelbrooklyn

      So another example that parentheses can change things is when you solve (8-3)x2 and 8-3x2.
      So, let's solve the first one. According to the order of operations, we have to do parentheses first. 8-3 is 5. Then we do the multiplication. 5 times 2 is 10 so the answer to the first problem is 10.
      Now let's solve the next one. Since there are no parentheses we solve the multiplication fist. 3 times 2 is 6. Then we do the subtraction. 8-6 is 2 so our answer is 2.
      In conclusion your answers can be different based on parentheses. I hope this helps!
      (1 vote)
  • blobby green style avatar for user justices4475
    Doesn't a negative plus a negative equal a positive number.
    (8 votes)
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    • hopper cool style avatar for user Philip
      A negative plus a negative will always equal a negative because you are taking a value less than zero and "applying more negativity to it" For example, you are already underground, and adding a negative distance means you will be digging even deeper underground.
      However, a negative times a negative will become positive.
      (16 votes)
  • marcimus orange style avatar for user elizabeth😻
    what if there is two perethisis and that's what I don't get and what I mean is witch one should I do first if you know lease explain it to me thanks so much!
    (2 votes)
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    • leaf red style avatar for user ENDER HΛVEN『Free coding advice』
      In the example equation:
      (10 × 4) + (4 ÷ 2)
      You must solve the parenthesis first (Moving left to right)
      (10 × 4 = 40) + (4 ÷ 2)
      40 + (4 ÷ 2 = 2)
      40 + 2 = 42

      In this other example parentheses inside of parentheses:
      ((10 × (4 ÷ 2)) × 2) + 2
      You solve the most nested parentheses first.
      ((10 × (4 ÷ 2 = 2)) × 2) + 2
      ((10 × 2 = 20) × 2) + 2
      (20 × 2 = 40) + 2
      40 + 2 = 42

      Did this help? If so, great! If not, explain to me what you still don't understand.
      (19 votes)
  • blobby green style avatar for user Anya
    1-(4-2x12)+7 in that problem,inside the parentheses, do i do the multiplication first, or the subtraction?
    (5 votes)
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  • duskpin seed style avatar for user 27knighti
    I have a Quick Questions how bo you do that
    (5 votes)
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  • duskpin seedling style avatar for user thegurleen09
    Isn't this basically Bidmas??
    (3 votes)
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  • hopper cool style avatar for user Gracie
    At , i really dont get it
    (2 votes)
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  • mr pink orange style avatar for user AMIYAHj
    For a company picnic, Todd fills 30 paper cups with ketchup and Yuki fills 27 paper cups with ketchup. Each paper cup holds 1 ounce. Later, Todd and Yuki learn that two other company employees completed exactly the same task.

    Question 1
    Part A

    Write a numerical expression to find how many ounces of ketchup the four employees prepared.

    Enter the correct answer in the box.
    (2 votes)
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Video transcript

What I want to do is think about whether this expression right over here would evaluate the same way whether or not we had parentheses. So to think about that, let's first think about how it would evaluate if we add the parentheses. So if we add the parentheses, we want to do what's ever in the parentheses first. And so here we have 8 minus 3, which is equal to 5. So this simplifies to 5 times 5 times 8 minus 3. And now we want to do the multiplication before we do subtraction. This goes back to order of operations. You do your multiplication and division first. Well, you do your parentheses first. Then if you have multiplication, division, addition, and subtraction all in a row, you want to do your multiplication and your division first. So here we're going to multiply 5 times 8 to get 40, and then we're going to subtract 3 to get 37. Now, let's think about what this would evaluate to if we did not have the parentheses. So it would be 8 minus 3 times 8 minus 3. So we just have to remind ourselves about the order of operations. The convention is to do your multiplication first. So you're actually going to multiply the 3 times the 8 before you subtract it from this 8 and then before you subtract this 3. So we took away the parentheses, but the order of operations say, hey, do this multiplication first. We could even put a parentheses here to emphasize that. So this will become 8 minus 8 minus 24. Let me write it this way. 8 minus 24 minus 3. 8 minus 24 minus 3. Now, 8 minus 24 is negative 16. You subtract another 3, you're going to get to negative 19. So clearly, you get very, very different values depending on whether or not you have parentheses.