Order of operations

There are four principal arithmetic operations addition, subtraction, multiplication, and division. When an equation involves multiplication and/or division 

Since this equation mixes addition and multiplication, we need to do the multiplication first  

Parentheses and division bars (fraction bars) are both symbols of enclosure. Whenever there are symbols of enclosure, execute any arithmetic operations inside the symbol of enclosure first. The general rule is to work from the innermost symbol of enclosure to the outermost, multiplying and dividing then adding and subtracting. Consider this example, which involves both kinds of symbols of enclosure and order of operations. 

To solve this example, we need to first evaluate inside the parentheses, then multiply and finally add  

Here we come to a very common error. Many students are tempted to divide 30 by 3 first, and then multiply by 5. But the division bar is a symbol of enclosure, so multiply 3 times 5 first and then divide 30 by 15. 

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Many runaway reactions can be prevented by changing the order of operations, reducing the temperature, or changing another parameter. 

Possible solutions include changing the oxidation level by FGl to (8) or reversing the order of operations so that R 0 acts as the nucleophile,... 

B. Handling Constraints on the Temporal Ordering of Operational Goals..337... 

Another important consideration regarding the character of constraints, h,, is their hierarchical articulation at multiple lerels of detail. For example, suppose that one is planning the routine startup procedure for a chemical plant. Safety considerations impose the following constraint on the temporal ordering of operations ... 

In this section we will present a formalized methodology that allows the transformation of quantitative bounding constraints into constraints on the temporal ordering of operators within the spirit of nonmonotonic planning. 

User-specified, temporal ordering of operational goals at higher levels of abstraction is propagated downwards in the hierarchy goals and is ultimately expressed as temporal ordering of primitive operations (see Section III,B). 

If the initial concentrations of the raw material were (in moles per liter), [HCNlo = 1, [H2S04]q = 1.5, and [HCHO]q = 2, then the preceding sequence of operations would immediately have violated Constraint-2, since [HCN] = 1 > 0.1. Similarly, Constraint-3 is violated by the preceding sequence, since [HSO4] = 0 < 1.5. In Section III,D, we discuss the notion of demotion of Clobberers. Let us see how it works here and leads to temporal ordering of operations ... 

The following three sequences satisfy all the preceding constraints on the temporal ordering of operations ... 

As in order of operations, evaluate the exponent first and then multiply -1 x (5 x 5 x... 

In a rough preliminary design for a waste treatment plant the cost of the components are as follows (in order of operation)... 

The basic operations of real numbers include addition, subtraction, multiplication, division, and exponentiation (discussed in Chapter 7 of this book). Often, in expressions, there are grouping symbols—usually shown as parentheses—which are used to make a mathematical statement clear. In math, there is a pre-defined order in which you perform operations. This agreed-upon order that must be used is known as the order of operations. 

In Chapter 2, the absolute value operation was reviewed. For order of operations, the absolute value symbol is treated at the same level as parentheses. 

For further information on the order of operations, refer to Practical Math Success in 20 Minutes a Day, Lesson 20, Miscellaneous Math, published by LearningExpress. 

If your calculator has parentheses keys, then it most likely will perform the correct order of operations. Check your calculator with these examples to see if it performs the correct order of operations. To evaluate 16 -100 5, enter 00000000 0. Your calculator should show a result of -4. 

Sometimes it is convenient to change the order of operations. The real numbers share some properties with which you should be familiar. These properties allow you to change the rules for the order of operations. They can be used to increase speed and accuracy when doing mental arithmetic. These properties are also used extensively in algebra, when solving equations. 

Recall that the order of operations directs you to add or multiply working from left to right. When you balance your checkbook, and have to add up a string of outstanding checks, list them all and use the commutative property to arrive at the total. Then, change the order of addends to add pairs whose unit (ones) digit adds to ten. 

The order of operations is to first evaluate parentheses, then exponents, then multiplication and division, left to right, and finally addition and subtraction, left to right. 

The order of operations can be remembered as PEMDAS, or Please Excuse My Dear Aunt Sally. 

The properties of numbers enable you to change the order of operations. 

The order of operations was covered in Chapter 3, Properties of Numbers. Be aware of some distinctions when working with the order of operations and exponents. Exponents are done after parentheses and before any other operations, including the negative sign. For example, -32 = -(3 x 3) = -9 because you first take the second power of three and then the answer is negative. However, (-3)2 = -3 x -3 = 9, since -3 is enclosed in parentheses. Following are some examples of order of operations with exponents. 

The order of operations directs multiplication to be done next, left to right. The answer will be left in terms of it. 

Order of operations directs multiplication to be done next, left to right. 

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