Section 1.2 Exponents and the Order of Operations

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1 4. ( (4 + 1) ( + ) ) = ( 5 ( 8 )) = (5 5) = ( -51 ) = -15 Technical Writing 44. Explain different methods for grouping expressions and why it s important to have them. The student s answer should discuss symbols such as parentheses, ( and ), brackets, { and } and brace, [ and ]. A complete answer would also discuss the division symbol. 45. A student claimed that (4 + 1) = 4 + 1? Explain the student s error. The correct approach here would be to start by adding the 4 and the 1 and then multiply by the. Section 1. Exponents and the Order of Operations In This Section We re going to cover two only slightly related topics in this section. The first of the two, exponents, is another example of the shorthand that we discussed in the previous section. They will give us a way to condense the multiplication of multiple numbers into a much more compact expression. The second topic, the order of operations, specifies the order in which operations like exponents, multiplication and addition have to be done to guarantee that every numeric expression simplifies to exactly one value. Learning Objectives 1. Define an integer exponent.. Simplifying numeric expressions using integer exponents.. Explain the purpose of the Order of Operations. 4. Use the Order of Operations to simplify numeric expressions. Required Material Negative Numbers In the previous section we reviewed negative numbers, including the process for multiplying them. In this section, we re going to be multiplying multiple numbers at a time, e.g To multiply multiple positive/negative numbers, you should take them in pairs. For example in , if you multiply together the first two - s, you ll get Now, take the next two numbers, 9 and -. If you multiply them together the expression becomes - - Now we re done to two numbers so the final result would be 81. The general rule is to multiply the numbers in the product two at a time until you get down to a single value. Here are some more examples for you to work through = 10-4 = 0-4 = = = -1 = = = = = 0 5

2 Teaching Suggestions Exponents Ask your students to write out some lengthy multiples, e.g. three times itself five times ( ) or eight times itself nine times. Talk about how difficult it would be to write out even bigger products like 10 times itself twenty times. This is the justification for exponents: 5 is much easier to write than. Order of Operations You can set the stage for the mathematical discussion by talking about non-mathematical procedures where the order you do the steps is important. Asks your students to come up with some examples. If they need one to get them started, suggest building a house. The plumbing has to be installed before the dry wall goes up and the dry wall has to be in place before the painters arrive. To transition to the mathematics discussion, ask your students to evaluate Some will do the multiplication first and end up with = 1 where others may do the addition first, giving them 4 4 = 1. Discuss how this is a problem. We need every expression, e.g , to mean one and only one thing. If it could mean multiple things, there would be constant confusion over what the person who wrote it meant. The order of operations solves this issue for us. By specifying which operations have a higher priority than others, it guarantees that every expression can be simplified in one and only one way. Common Errors Acronyms There s a popular acronym that students can use to remember the right order: PEMDAS. The letters stand for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction but Please Excuse My Dear Aunt Sally is a little easier to remember. That s a useful way to remember the order but I ve seen it cause confusion. It makes it look like multiplication has a higher priority than division when that isn t the case. You should remind your students that multiplication and division are on the same level and should always be done from left to right. For example 1 / 4 should be simplified doing the operations from left to right: 1 / 4 = = 9 Additional Examples Evaluate the following exponential expressions = = = -( ) = -( ) = - Sometimes students will see the M before the D in PEMDAS and (incorrectly) think that means multiplication comes first. Doing the simplification that way would give an incorrect value: incorrect: 1 / 4 = 1 / 1 = = = = -(4 4 ) = -( ) = -5

3 = =.001. (-4) (-4) = -4-4 = Simplify the following expressions = 1 = -10 The Order of Operations specifies that the multiplication has to be done first. 11. / 5 / 5 = 5 = 10 Multiplication and division have to be simplified from left to right which, in this case, means the division comes first ( 4) 5 + ( 4) = 5 + () = = 9 What s inside parentheses has to be simplified first. Exercises Solutions = 5 1 = -11 The Order of Operations specifies that the exponent has to be done first / 4 / = 4 1 = Division comes first and then subtraction. Evaluate the following exponential expressions. 1. = = = = ( + 1 ) + 10 / ( + 1 ) + 10 / = ( + 1) + 10/ = / = = = = = - ( ) = = = (.1) =.1.1 =.01 You have to start by simplifying the parentheses and, inside the parentheses, the exponent comes first.. - (-) = -( ) (- -) = -4 4 = = = 9. / = ( ) ( ) / ( ) = / = Simplify the following expressions = 5 = = = = ( 5) / = - / = ( ) = 4 = = = -(5 5) = (1 4 / ) = + 4(1 ) = = 4 = 19. (5 ) + 4 ( 1) = (5 ) + 4( 1) = (5 4) + 4 = = + 8 = 11

4 0. [4 ( + 4 )] / = [4 ( + )] / = [4 (9 + 1)] / = [4 5] / = [4 50] / = -4 / = -18 / = / 4 = = 9. [11 + ( 5) ] + 1 / = [11 + (-)] + 4 = [ ] + 4 = = 4. (5 1) + (1 5) = 4 + (-4) = = = 4. (-5) 5 = = 5 5 = (4 - ) (4 4 ) (1 ) ( - ) 9( ) 9(8 4) 9 4. {5 + ( / ) } + 5 = {5 5 + ( ) } + 5 = {5 + (4)} + 5 = { } + 5 = = ( + 1) [5 + 1] = 5 ( + 1) [ ] = 5 ( + 4 1) [5 + 1] = 5 ( 1) = 5 8 = 0 8 = 10 8 = -8 Calculate the following values. 9. Calculate the area of a rectangle whose sides are 19 cm by cm. 19 cm cm = 418 cm 0. Calculate the area of a rectangle whose sides are.5 in. by 1 in..5 in 1 in = 0 in 1. Calculate the area of a square whose sides are all 5 ft. 5 ft 5 ft = 5 ft. If a table is 5 by, what s the minimum size of a tablecloth that will cover it? 5 = 15 sq. ft.. The top of a box is by 8. The sides are by and 8 x. How much wrapping paper will be required to just cover all the sides? The area of the top is 8 = 4 sq. in. The area of the first side is = sq. in.; the area of the second side is 8 = 1 sq. in. Since there are two of each side and the top and bottom have the same dimensions, the total area is Technical Writing = 9 sq. in. 4. Describe what it means for one number to be raised to another. Raised is another way of describing an exponent. For example, three raised to the fifth power is the same as Describe the difference between - and (-). The difference is in how the negative sign is handled. In the first expression, only the is being squared, not the negative sign. In the second expression, the whole - is being squared. The parentheses are required to make the negative sign part of what is being raised to the exponent.. Why do we need an order of operations? We need an order of operations to make sure that every expression can be simplified in exactly one way. Without it, one expression could simplify to multiply numbers. Investigations. Write 4 as a single exponent. What relationship do you see between the exponents in the original version and your answer? 4 = ( ) ( ) = = Notice that if you add the two original exponents, you get the exponent of the simplified version. That s a relationship that we ll talk about in more detail in algebra two. 8

5 8. Write as a single exponent. What relationship do you see between the exponents in the original version and your answer? the two original exponents, i.e. = 4. 4 In this example, the new exponent was the difference of Find the Error Explain why each of the following statements is incorrect and correct it = 9 = / = 4 4 = 1 Multiplication has to be simplified before addition. The correct simplification would be + = + 1 = = 15 It looks like the exponent was treated as multiplication, i.e. 5, rather than the number of times to multiply the. The correct result would be = = = = 4 The exponent only applies to the thing immediately in front of it. In this case, that means the 10, not the -. That makes the expression: -10 = -(10 ) = -(10 10) = -100 Multiplication and division have to be simplified from left to right. In this case, that means the multiplication has to be done first: 4 1 / = 48 / = = = 9 Exponents have to be simplified before addition. That makes this expression equal to = = = 5 Multiplication has to be simplified before addition. The correct simplification would be + 8 = = Section 1. Graphing Numbers In This Section In Section 1.1, we discussed translating English into mathematics. In this section, we re going to make a different kind of translation from English/mathematics into pictures. Then we ll see how our new translation applies directly to the real world by equating it with distance and measurement. Most people are visually oriented. We understand best when we see something rather than hearing about it. This is especially true with mathematics. Whenever possible I encourage you to try to draw pictures that illustrate ideas, especially with things that you find confusing. In this section, we do this by placing numbers on a line and equating their position relative to each other with distance. Learning Objectives 1. Construct a number line. Locate numbers on a number line.. Calculate the distance between points on a number line. 9