Order of Operations More Essential Practice


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1 Order of Operations More Essential Practice We will be simplifying expressions using the order of operations in this section. Automatic Skill: Order of operations needs to become an automatic skill. Failure to master this concept without a calculator will make success in algebra very difficult, if not impossible. Order of Operations: Order of operations tell us what to do first in a complicated expression. For example, would we add or multiply first with + 3 * 3? By order of operations, we multiply first and our answer is 11. Here are the order of operations: 1. Parenthesis: Work everything inside parentheses first. Parentheses are grouping symbols. If there are nested groupings, you may also see brackets and braces: {1 + [34( + 1)]}. Brackets and braces are just like parentheses and are used to help make the groupings clearer. Always work from inside out when there are nested groupings. See example 1. There are understood parentheses in problems as well. They are left out because it would look too clunky to have them. The main places we will see understood parentheses will be with fractions (numerator and denominator) and with radicals (5 3) (radicand). Examples: (5 7) is clunky and will be written as ; (5 16) is Courtesy mmarden 1
2 clunky and will be written as It may be tempting to simplify by dividing out the 5 s. But this would violate order of operations since there are understood parentheses around the numerator and denominator. Similarly, with 5 16 it is tempting to think we can take the square root of 5 and of 16 and then subtract, but we can not. Again this would be violating the order of operations since there are understood parentheses around the radicand (ie, 516) and the answers would be different! Sometimes parentheses are used to indicate multiplication and not a grouping. If we have 5 (3)(), there is nothing to do inside either of the parentheses. This means that they are NOT present to indicate a grouping, but to indicate that 3 and are to be multiplied.. Exponents: Expand exponents after groupings and before other operations. It is important to note the difference between 3 and 3. When we have 3, we are only squaring the 3, not the negative! Another way to write this would be 33. The result is 9. However, 3 means that the entire number, which is negative three, is to be squared. Another way to write this would be 3 3. The result is Multiplication and Division: Perform multiplication and division in the order it appears from left to right after exponents are expanded. If we have 8 4 we would first divide 8 by (final result is 16). If we have 8 4we would first Courtesy mmarden
3 multiply 8 and (final result is 4). Often there are assumed multiplication in a problem. If we have 3(5 1), don t add the 3 and! There is an assumed multiplication between the and what is in parentheses. Final result is 11. The reason why we do multiplication and division in order from left to right is because divisions can be written as multiplication of the reciprocal. So, 8 4 is the same as When in this form, we see that no matter what multiplication is done first, we have a 16 for the result. 4. Addition and Subtraction: Perform addition and subtraction in the order it appears from left to right after all multiplications and divisions are completed. If we have 6 1 we would first subtract 6 from (final result is 3). If we have 6 1 we would first add and 6 (final result is 7). Similarly to multiplication and division, the reason why we do addition and subtraction in order from left to right is because subtractions can be written as additions. So, 6 1 is the same as done first, we have 3 for the result Here no matter which addition is Ways to remember the order of operations: 1. PEMDAS: The letters stand for the operations (parenthesis, exponents, multiplication, division, addition, subtraction).. Please Excuse My Dear Aunt Sally: The first letter in each word stands for the operations (parenthesis, exponents, multiplication, division, addition, subtraction). Courtesy mmarden 3
4 Each of these will help you remember the general order, BUT do not forget that multiplication and division as well as addition and subtraction are performed in the order they appear from left to right!!! Comment on how you write your work: Mathematics involves communication of ideas through symbols (numbers: 5 instead of five ; operations: + instead of sum; etc...). It is very important to learn how to write your work in a way that communicates to others what you did. Mathematics is often called the universal language as often people can understand math notation when they can not speak a common language. Search for alien life often includes mathematics for the same reason! But, it isn t universal if your work is not presented in an orderly fashion using equal marks appropriately because others may not be able to follow it. At PCC it is a departmental policy that the write up of your math be a part of your grade. With an online class it is more difficult to for me to correct mistakes in presentation before the campus exams. Please see Presentation Requirements for written work. Also pay attention to how the examples are presented in these lecture notes and comments regarding use of parentheses and such. If you learn to write your work correctly, you will eventually see fewer mistakes and your road to higher mathematics will be easier. Presentation Requirements for Expressions: For expressions the write up should have equals with each step to indicate that the expression in each step is equivalent to the expression in the prior step. We first rewrite the original problem and put the first step on the top line with equals between them. Then continue each step in a vertical format lining up the equals on the left hand side as shown below. It is VERY important to recopy parts of the problem that are not altered. Reasons: 1. It is incorrect to have equals when there are missing parts to the problem; Courtesy mmarden 4
5 . I can not determine what has changed easily (and will not spend much time trying to) 3. Often students forget pieces of the problem and end up with the wrong answer. Here is the format: Original Problem = First Step = Next Step = Continue with all needed steps = Last Step For online discussion boards: It is very difficult to align equals on discussion boards. For this reason, the following vertical format is acceptable and preferred (but only for discussion boards): Original Problem = First Step = Next Step = Continue with all needed steps = Last Step You may not put in every step that I did and you may work problems differently, but the structure should be similar: vertical format, correct notation, appropriate use of equals signs, and easily followed from one step to the next! Very Important Reminder on Calculators: You will not have access to a calculator on campus tests (the major part of your grade) except for word problems. Work examples and homework without a calculator to ensure that you have the arithmetic skills needed to be successful on the campus tests. Courtesy mmarden 5
6 EXAMPLE 1: Simplify using the order of operations. a) {1 + [34( + 1)]} b) a) You may replace brackets and braces with parentheses if you wish. Here I kept the brackets and braces instead of replacing them with parentheses (either is fine). Just be careful to not drop grouping symbols until everything is done within the grouping. Note that each part of the problem that has not been changed is recopied. Also note how the first line contains the original problem and the first step separated by equals and then the remaining steps are lined up vertically b) Note how each part of the problem that hasn t been altered is rewritten and carried to the next step. This will be expected in your work. Also note that when parts of the problem are separated by groupings or by + or , it is ok to work the parts separately since doing so won t violate the order of operations. For example: The 5 was squared in the denominator and the numerator in the second fraction was combined before the grouping parentheses were cleared in the numerator of the first fraction. It is fine to do this, just be very careful! Courtesy mmarden 6
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