course MTH 163 ????|?wq??{[?[?{assignment #001
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03:32:04 `q001. Explain the difference between x - 2 / x + 4 and (x - 2) / (x + 4). The evaluate each expression for x = 2.
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RESPONSE --> In the first equation, x-2/x+4, the term 2/x would be solved first, per descending order of power, and would then be subtracted from x and added to 4. In the second equation, (x-2)/(x+4), each equation in parenthesis would be solved indivually, then divided. In the first equation I would substitute x for 2, giving me 2-2/2+4. I would then solve 2/2, giving me 2-1+4. I could then solve either the subtraction or addition either way. 2-1+4 = 1+4 = 5. In the second equation I would substitute x for 2, giving me (2-2)/(2+4). I would then solve each term in parens seperately, giving me 0/8. Of course, 0 divided by anything is 0. confidence assessment: 2
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03:34:58 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) / ( 4 - 2) = 0 / 2 = 0.
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RESPONSE --> I must've misread the second equation. I thought it was (x+4). Of course, I still the correct answer of 0, but still misread it and as such did the wrong math. self critique assessment: 1
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03:42:42 `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.
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RESPONSE --> In the first expression, 2 is raised to the x power. The result of that is then added to 4. In the second equation 2 is raised to the power of the result given by (x+4). In the first expression I'd first substitute x for 2, giving me 2^2+4. Taking care of the exponents first, I'd raise 2 to the 2 power, yielding 4. I'd then add that to 4, giving me an answer of 8. In the second expression, I'd first substitute x for 2, giving me 2^(2+4). Taking care of the parens first, I'd add 2 to 4, giving me 2^6. I'd then solve the exponential, giving me an answer of 64. confidence assessment: 2
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03:44:12 2 ^ 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.
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RESPONSE --> self critique assessment: 3
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04:01:00 `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?
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RESPONSE --> The numerator is 3. The denominator is [(2x-5)^2*3x+1]. First I substitute x for 2, giving me 2-3/[(2*2-5)^2*3*2+1]-2+7*2. Next I take care of parens and brackets, giving me 2-3/[(4-5)^2*3*2+1]-2+7*2 = 2-3/[(-1)^2*3*2+1]-2+7*2. Next I'd solve exponents, multiplications and division, then addition and subtraction, in that order, WITHIN the brackets and parens. 2-3/[1*3*2+1]-2+7*2 = 2-3/[3*2+1]-2+7*2 = 2-3/[6+1]-2+7*2 = 2-3/7-2+7*2. Now I solve first the multiplication and division, then addition and subtraction. 2-3/7-2+7*2 = 2-3/7-2+7*2 Now with the parens and brackets eliminated I solve per order of descending power. 2-3/7-2+7*2 = 2-0.43-2+14 = 1.57-2+14 = -0.43+14 = 13.57 confidence assessment: 1
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04:02:45 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 ].
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RESPONSE --> self critique assessment: 3
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04:06:59 `q004. Explain, step by step, how you evaluate the expression (x - 5) ^ 2x-1 + 3 / x-2 for x = 4.
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RESPONSE --> First we substitute x for 4. (4-5)^(2*4)-1+3/4-2 We then evaluate all parens. (4-5)^(2*4)-1+3/4-2 = -1^8-1+3/4-2 We then evaluate exponents. -1^8-1+3/4-2 = 1-1+3/4-2 We then we perform multiplication and division. 1-1+3/4-2 = 1-1+0.75-2 Then we add and subtract. 1-1+0.75-2 = 0.75-2 = -1.25 confidence assessment: 2
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04:10:26 We get (4-5)^2 * 4 - 1 + 3 / 1 - 4 = (-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). COMMON ERROR: (4 - 5) ^ 2*4 - 1 + 3 / 4 - 2 = -1 ^ 2*4 - 1 + 3 / 4-2 = -1 ^ 8 -1 + 3 / 4 - 2. INSTRUCTOR COMMENTS: There are two errors here. In the second step you can't multiply 2 * 4 because you have (-1)^2, which must be done first. Exponentiation precedes multiplication. Also it isn't quite correct to write -1^2*4 at the beginning of the second step. If you were supposed to multiply 2 * 4 the expression would be (-1)^(2 * 4). Note also that the -1 needs to be grouped because the entire expression (-1) is taken to the power. -1^8 would be -1 because you would raise 1 to the power 8 before applying the - sign, which is effectively a multiplication by -1.
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RESPONSE --> Per the instructor comments at the bottom I made the mistake of multiplying the 2 by the 4 before raising the -1 to the 2. I see where I made my mistake and understand what I should've done. self critique assessment: 1
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04:12:01 *&*& Standard mathematics notation is easier to see. On the other hand 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 have to enter expressions through a typewriter, so it is well worth the trouble to get used to this notation. Indicate your understanding of the necessity to understand this notation.
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RESPONSE --> Standard notation is easier to understand. It should be used when making notes and solving problems on paper, then converted back to competer algebra when typed in. self critique assessment: 2
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04:14:56 `q005. At the link http://www.vhcc.edu/dsmith/genInfo/introductory problems/typewriter_notation_examples_with_links.htm (copy this path into the Address box of your Internet browser; alternatively use the path http://vhmthphy.vhcc.edu/ > General Information > Startup and Orientation (either scroll to bottom of page or click on Links to Supplemental Sites) > typewriter notation examples and you will find a page containing a number of additional exercises and/or examples of typewriter notation.Locate this site, click on a few of the links, and describe what you see there.
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RESPONSE --> Each link that I've clicked on shows the expressions accompanying the link in standard notation. confidence assessment: 3
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04:15:33 You 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,
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RESPONSE --> self critique assessment: 3
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04:15:45 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.
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RESPONSE --> self critique assessment: 3
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04:15:52 end program
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RESPONSE --> self critique assessment: 3
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