#$&* course mth 151 1/16/2014 3:14 pmI don't have a text editor such as Notepad. I only Microsoft word so it previously came out as come out a massive paragraph. I apologize for that. I am not sure how to change this because when I copy and paste it in the box below, the spaces are there.
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Given Solution: `a The order of operations dictates that grouped expressions must be evaluated first, that exponentiation must be done before multiplication or division, which must be done before addition or subtraction. It makes a big difference whether you subtract the 2 from the x or divide the -2 by 4 first. If there are no parentheses you have to divide before you subtract. Substituting 2 for x we get 2 - 2 / 2 + 4 = 2 - 1 + 4 (do multiplications and divisions before additions and subtractions) = 5 (add and subtract in indicated order) If there are parentheses you evaluate the grouped expressions first: (x - 2) / (x + 4) = (2 - 2) / ( 2 + 4 ) = 0 / 6 = 0. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ok ********************************************* Question: `q002. Explain the difference between 2 ^ x + 4 and 2 ^ (x + 4). Then evaluate each expression for x = 2. Note that a ^ b means to raise a to the b power. This process is called exponentiation, and the ^ symbol is used on most calculators, and in most computer algebra systems, to represent exponentiation. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 2 ^ (x + 4) 2^ x + 4 2^ (2 + 4) 2^ 2 + 4 2^ 6 2*2 + 4 2*2*2*2*2*2 = 64 4 + 4 = 8 confidence rating #$&*: I am confident in my ability to do this problem as well ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a2 ^ x + 4 indicates that you are to raise 2 to the x power before adding the 4. 2 ^ (x + 4) indicates that you are to first evaluate x + 4, then raise 2 to this power. If x = 2, then 2 ^ x + 4 = 2 ^ 2 + 4 = 2 * 2 + 4 = 4 + 4 = 8. and 2 ^ (x + 4) = 2 ^ (2 + 4) = 2 ^ 6 = 2*2*2*2*2*2 = 64. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ok ********************************************* Question: `q003. What is the numerator of the fraction in the expression x - 3 / [ (2x-5)^2 * 3x + 1 ] - 2 + 7x? What is the denominator? What do you get when you evaluate the expression for x = 2? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Numerator = 3 Denominator = [ ( 2x - 5) ^2 * 3x + 1] x - 3 / [ (2x-5)^2 * 3x + 1 ] - 2 + 7x 2 - 3 / [ ( 2*2 - 5)^2 * 3*2 +1] -2 + 7*2 2 - 3 / [ (4 -5) ^2 * 6 + 1] - 2 + 14 2 - 3 / [ (-1) ^2 * 6 + 1] - 2 + 14 2 - 3 / [ 1 * 6 + 1] - 2 + 14 2 - 3 / 7 - 2 + 14 2 - 2 + 14 - 3 / 7 = 14 - 3/7 is it necessary to rearrange this step or is it just easier this way? 98/7 - 3/7 = 95/7 confidence rating #$&*: I am horrible with algebraic fractions. Having said that and read the given solution, I can follow your reasoning but it’s still slow going. I did a step, then checked my answer against yours until I began to understand what I was seeing. Given time, I believe I can pick this back up a tad quicker ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a: The numerator is 3. x isn't part of the fraction. / indicates division, which must always precede subtraction. Only the 3 is divided by [ (2x-5)^2 * 3x + 1 ] and only [ (2x-5)^2 * 3x + 1 ] divides 3. If we mean (x - 3) / [ (2x-5)^2 * 3x + 1 ] - 2 + 7x we have to write it that way. The preceding comments show that the denominator is [ (2x-5)^2 * 3x + 1 ] Evaluating the expression for x = 2: - 3 / [ (2 * 2 - 5)^2 * 3(2) + 1 ] - 2 + 7*2 = 2 - 3 / [ (4 - 5)^2 * 6 + 1 ] - 2 + 14 = evaluate in parenthese; do multiplications outside parentheses 2 - 3 / [ (-1)^2 * 6 + 1 ] -2 + 14 = add inside parentheses 2 - 3 / [ 1 * 6 + 1 ] - 2 + 14 = exponentiate in bracketed term; 2 - 3 / 7 - 2 + 14 = evaluate in brackets 13 4/7 or 95/7 or about 13.57 add and subtract in order. The details of the calculation 2 - 3 / 7 - 2 + 14: Since multiplication precedes addition or subtraction the 3/7 must be done first, making 3/7 a fraction. Changing the order of the terms we have 2 - 2 + 14 - 3 / 7 = 14 - 3/7 = 98/7 - 3/7 = 95/7. COMMON STUDENT QUESTION: ok, I dont understand why x isnt part of the fraction? And I dont understand why only the brackets are divided by 3..why not the rest of the equation? INSTRUCTOR RESPONSE: Different situations give us different algebraic expressions; the situation dictates the form of the expression. If the above expression was was written otherwise it would be a completely different expression and most likely give you a different result when you substitute. If we intended the numerator to be x - 3 then the expression would be written (x - 3) / [(2x-5)^2 * 3x + 1 ] - 2 + 7x, with the x - 3 grouped. If we intended the numerator to be the entire expression after the / the expression would be written x - 3 / [(2x-5)^2 * 3x + 1 - 2 + 7x ]. STUDENT COMMENT: I wasn't sure if the numerator would be 3 or -3. or is the subtraction sign just that a sign in this case? INSTRUCTOR RESPONSE: In this case you would regard the - sign as an operation to be performed between the value of x and the value of the fraction, rather than as part of the numerator. That is, you would regard x - 3 / [ (2x-5)^2 * 3x + 1 ] as a subtraction of the fraction 3 / [ (2x-5)^2 * 3x + 1 ] from the term x. STUDENT QUESTION: There was another question I had about this problem that wasn’t addressed. At the end when you changed the order of operation from 2 - 2 + 14 - 3/7 = 14 - 3/7 where did the 98/7 - 3/7 come into play before the end solution of 95/7? I must have forgotten how to do this part. INSTRUCTOR RESPONSE: It's not clear how you can get 95/7 without this step. To do the subtraction 14 - 3/7 both terms must be expressed in terms of a common denominator. The most convenient common denominator is 7. So 14 must be expressed with denominator 7. This is accomplished by multiplying 14 by 7 / 7, obtaining 14 * 7 / 7 = 98 / 7. Since 7/7 = 1, we have just multiplied 14 by 1. We chose to use 7 / 7 in order to give us the desired denominator 7. Thus our subtraction is 14 - 3/7 = 98/7 - 3/7 = (98 - 3) / 7 = 95 /7. STUDENT COMMENT It took me a while to think thru this one especially when I got to working with the fraction. Fractions have always been my weak spot. Any tips to make working with fractions a little easier is greatly appreciated. INSTRUCTOR RESPONSE Fractions are seriously undertaught in our schools, so your comment is not unusual. I have to focus my attention on the subject matter of my courses, and while I do address it to a point, I don't have time to do justice to the subject of fractions. In any case , to do so would be redundant on my part, since there are a lot of excellent resources on the Internet. I suggest you search the Web using something like 'review of fractions', and find something appropriate to your needs. You should definitely review the topic, as should 95% of all students entering your course. STUDENT COMMENT I think I am confused on why the Numerator is not the top portion and denominator the bottom portion of the problem. INSTRUCTOR RESPONSE Everything is on one line so there is no top or bottom in the given expression. A numerator and denominator are determined by a division of two expressions. As we know, a denominator divides a numerator. In the given expression the division sign occurs between the 3 and the [ (2x-5)^2 * 3x + 1 ], so 3 is the numerator and [ (2x-5)^2 * 3x + 1 ] is the denominator. x is not divided by the denominator, since the division occurs before the subtraction. For the same reason the -2 + 7x is not involved in the division. So neither the x nor the -2 + 7 x is part of the fractional expression. STUDENT COMMENT Didn’t know that 3 / 7 was 3/7 as a fraction. INSTRUCTOR RESPONSE 3/7 is treated as a fraction because of the order of operations. 3 must be divided by 7 before any other operation is applied to either number, and 3 divided by 7 is the fraction 3/7. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: 2 ********************************************* Question: `q004. Explain, step by step, how you evaluate the expression (x - 5) ^ 2x-1 + 3 / x-2 for x = 4. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (x - 5) ^ 2x-1 + 3 / x-2 (4 - 5) ^ 2*4 -1 + 3 / 4 -2 (-1) ^ 2 * 4 -1 + 3 / 4 -2 1 * 4 -1 + 3 / 4 -2 4 -1 +3/4 -2 4 - 1 - 2 + 3/4 1+ ¾ = 7/4 confidence rating #$&*: I still step checked my answer against yours, but this time the problem solving was a bit easier. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aWe get (4-5)^2 * 4 - 1 + 3 / 4 - 2 = (-1)^2 * 4 - 1 + 3 / 4 - 2 evaluating the term in parentheses = 1 * 4 - 1 + 3 / 4 - 2 exponentiating (2 is the exponent, which is applied to -1 rather than multiplying the 2 by 4 = 4 - 1 + 3/4 - 2 noting that 3/4 is a fraction and adding and subtracting in order we get = 1 3/4 = 7 /4 (Note that we could group the expression as 4 - 1 - 2 + 3/4 = 1 + 3/4 = 1 3/4 = 7/4). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: 2 ********************************************* Question: `q005. Evaluate the expression x^3x+2/x-1 for x = 2, according to the order of operations. Show all your steps. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x^3x+2/x-1 2^3 * 2 + 2/ 2 - 1 You do all the exponentials first 8 * 2 + 2/2 -1 Then follow the order of operations, division and multiplying first 16 +2 + 1 - 1 = 18 Then all adding and subtracting
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Given Solution: `aYou should see a brief set of instructions and over 30 numbered examples. If you click on the word Picture you will see the standard-notation format of the expression. The link entitled Examples and Pictures, located in the initial instructions, shows all the examples and pictures without requiring you to click on the links. There is also a file which includes explanations. The instructions include a note indicating that Liberal Arts Mathematics students don't need a deep understanding of the notation, Mth 173-4 and University Physics students need a very good understanding, while students in other courses should understand the notation and should understand the more basic simplifications. There is also a link to a page with pictures only, to provide the opportunity to translated standard notation into typewriter notation. end program STUDENT COMMENT (not quite correct) I see a collection of typewriter problems, after looking at some of them I see that the slash mark is to create a fraction rather than to denote division. INSTRUCTOR CORRECTION A fraction is a division of the numerator by the denominator. The slash mark indicates division, which can often be denoted by a fraction. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: 0k ********************************************* Question: `q007. Standard mathematics notation is easier to look at; it's easier to see the meaning of the expressions. However it's very important to understand order of operations, and students do get used to this way of doing it. You should of course write everything out in standard notation when you work it on paper. It is likely that you will at some point use a computer algebra system, and when you do you will probably have to enter expressions using a keyboard, so it is well worth the trouble to get used to this notation. As one example take a minute and go to Wolfram Alpha at http://www.wolframalpha.com/. If this link doesn't work just search the Web for 'Wolfram Alpha'. When the page comes up, you can simply copy the expression x - 3 / (2x + 4) into the box. Think about what you would get were you to evaluate this expression, then click on the = sign. Repeat the process with each of the following expressions. Be sure you think in each case about what expression you would expect to see. (x - 3) / (2 x + 4) x - 2 / 3 (x - 2) / 3 (x+2) ^ 2x (x+2) ^ (2x) (x - 3) / 3x (x - 3) / (3 x) x - 3 / 3x Do these expressions act the way most people would expect, or do they act in the way dictated by the order of operations? Indicate your understanding of why it is important to understand this notation. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I don’t see them acting in the normal order of operations, but I do like that wolframa does give you step by step to each equation, even if I don’t quite understand all the different equations presented. confidence rating #$&*: I feel good about this website " ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q007. Standard mathematics notation is easier to look at; it's easier to see the meaning of the expressions. However it's very important to understand order of operations, and students do get used to this way of doing it. You should of course write everything out in standard notation when you work it on paper. It is likely that you will at some point use a computer algebra system, and when you do you will probably have to enter expressions using a keyboard, so it is well worth the trouble to get used to this notation. As one example take a minute and go to Wolfram Alpha at http://www.wolframalpha.com/. If this link doesn't work just search the Web for 'Wolfram Alpha'. When the page comes up, you can simply copy the expression x - 3 / (2x + 4) into the box. Think about what you would get were you to evaluate this expression, then click on the = sign. Repeat the process with each of the following expressions. Be sure you think in each case about what expression you would expect to see. (x - 3) / (2 x + 4) x - 2 / 3 (x - 2) / 3 (x+2) ^ 2x (x+2) ^ (2x) (x - 3) / 3x (x - 3) / (3 x) x - 3 / 3x Do these expressions act the way most people would expect, or do they act in the way dictated by the order of operations? Indicate your understanding of why it is important to understand this notation. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I don’t see them acting in the normal order of operations, but I do like that wolframa does give you step by step to each equation, even if I don’t quite understand all the different equations presented. confidence rating #$&*: I feel good about this website " ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!