Question 536173: Simplify 16/4+{5×6[3+(4+1)]}
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Given to simplify: 16/4+{5×6[3+(4+1)]}
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A little acronym to give you some help about how to do problems such as this is PEMDAS. Remember is by using the first letters of "Please Excuse My Dear Aunt Susan". It stands for this: P = parentheses; E = exponents; M = multiply and D = Divide; and finally A = add and S = subtract. They are done in the order listed as you work from left to right. Clear parentheses first. Then take care of exponents. For the next step remember that Multiply and Divide are done together as you encounter them going left to right. Once you have done all those, then you do the same with Addition and Subtraction. They are done in the order you encounter them going from left to right.
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As you will see below, the answer to the problem you were given is 244. Let's now work it out.
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This problem has parentheses, so we work with those first. Go to the most interior set of parentheses and work with the contents of that set of parentheses first using E M D A and S in that order on the contents of the parentheses. The most interior set of parentheses is (4+1). It contains no exponents, multiplications or divisions to do. We just do the addition (there also are no subtractions to worry about.) The addition of 4+1 = 5 and we can replace (4+1) with (5) to get:
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16/4+{5×6[3+(5)]}
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Since the (5) is preceded by a + sign, we can just remove the parentheses and the problem becomes:
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16/4+{5×6[3+5]}
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Now the most interior set of the nested parentheses (or the brackets used in this problem) is [3+5]. So we work on the contents of those brackets. No exponents, no multiplications, no divisions, just an addition, and no subtractions. We do the addition of 3+5 to get 8 and we can then replace the [3+5] with [8]. The result is:
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16/4+{5×6[8]}
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There is an implied multiplication sign between the 6 and the [8]. We can remove the brackets and just insert that multiplication sign to get:
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16/4+{5×6×8}
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Now we have only one remaining set of parentheses/brackets. That set is:
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{5×6×8}
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The contents are done in this order: as you move from left to right, do the multiplication of 5 times 6 to get 30 and then do the 30 times 8 to get {240}. This makes the problem become:
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16/4+{240}
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Again note that the parentheses/brackets are preceded by a plus sign. This means that they can be removed with no change to get:
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16/4 + 240
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This problem is considerably simplified. There are no longer any parentheses to worry about. There never were any exponents to worry about either. What do we do now? We read the problem from left to right and do the operations of Multiplication and Division as the signs are encountered. (There is only a division to do.) Once we have all those done, we return and again read the problem from left to right and do the additions and subtractions as they are encountered. Here we go. Reading from the left the first sign encountered is the division sign. Do the division of 16 by 4 to get 4. This makes the problem become:
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4 + 240
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All the multiplications and divisions are now done. Read the remaining problem from left to right and the first sign is the plus sign. Add 4 to 240 and you get the final answer of 244.
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That's all there was to it. This one was pretty simple. In the way of an example for practice let's try one a little more difficult. Here's a problem I made up:
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15/3*2/10+5*4^2-(7*4/14-3+6/3)
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How would you do that using PEMDAS? First, go to the parentheses and read the contents from left to right. No exponents in it so do the M and D in the order encountered. We are only working on the contents of the parentheses. From left to right we first see the multiplication of 7*4 we do that and get 28. Next we see a division sign so we divide the 28 by 14 to get 2. Next we see a minus sign. Skip that, were only working on multiplication and division. Next we see a plus sign. Skip that. Finally we get to a division sign between the 6 and the 3. Divide 6 by 3 and get 2. Replace 6/3 with 2. What we now have is:
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15/3*2/10+5*4^2-(2-3+2)
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Within the parentheses the first 2 is what we got from multiplying 7 times 4 and then dividing that result by 14. The second 2 came from dividing 6 by 3.
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Now within the parentheses we read from left to right and we see 2-3. Subtract 3 from 2 to get -1. As a result we now have -1+2 in the parentheses. Add 2 to -1 and you have +1 in the parentheses. So the problem now is reduced to:
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15/3*2/10+5*4^2-(+1)
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The parentheses are preceded by a minus sign, so they can be removed by changing the sign of the term inside to get:
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15/3*2/10+5*4^2-1
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What next? No parentheses any more. But in PEMDAS order there is an exponent to take care of. Square the 4 to get 16 and substitute that into the problem:
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15/3*2/10+5*16-1
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We're done with parentheses and exponents. Now we read the problem from left to right and do the multiplications and divisions in the order that they are encountered:
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First divide the 15 by 3 to get 5. Then multiply that 5 times the 2 to get 10. Then divide that 10 by 10 to get 1. So far we can replace everything up to the plus sign with 1. This makes the problem:
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1+5*16-1
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Continue reading from left to right. Remember we're working on multiplications and divisions. So we skip over the + sign between the 1 and the 5 to get to the 5 times 16. The 5 times 16 is 80. Substitute that and the problem is now:
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1+80-1
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In PEMDAS we have now completed the PEMD actions. For the additions and subtractions (the AS) from left to right we now add 1 to 80 and get 81 and then we substract the last 1 to get 80. That's the answer to this made up problem. This problem was just for practice. It might help you to understand the order of operations a little better.
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Hope this helps.
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