course Phy 201 I am submitting this for both Phy 201 and Mth 163. ܊bxwܘ^~vassignment #001
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10:23:51 `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 --> The order of operations is different. In the 1st problem you divide first. In the 2nd problem you do the math in parenthesis before dividing. x - 2 / x + 4 2 - 2 / 2 + 4 2 - 1 + 4 1 + 4 = 5 (x - 2) / (x + 4) (2 - 2) / (2 + 4) 0 / 6 = 0 confidence assessment: 2
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10:28:48 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 wrote the second problem down as (x-2) / (x+4) which accounts for the difference in the last problem. Otherwise, I understand the order of operations. I wasn't sure whether you subtract or add first in the problem 2-1+4, but now I understand that you should add and subtract the way the problem is written. self critique assessment: 2
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10:32:43 `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 1st problem, 2 is raised to the 2nd power and then 4 is added. In the 2nd problem, 2 is rasied to the 6th because x+4 is in parenthesis. 2 ^ x + 4 2 ^ 2 + 4 4 + 4 = 8 2 ^ (x + 4) 2 ^ (6) = 64 confidence assessment: 2
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10:33:54 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 --> I understand that they are different because of the parenthesis. self critique assessment: 2
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10:54:04 `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 --> Numerator: x - 3 Denominator: [ (2x-5)^2 * 3x + 1 ] x - 3 / [ (2x-5)^2 * 3x + 1 ] - 2 + 7x 2 - 3 / ( (4-5)^2 * 6 + 1) - 2 + 14 -1 / (-1^2 * 6 + 1) - 2 + 14 -1 / (1 * 6 + 1) - 2 + 14 -1 / 7 - 2 + 14 -.14 - 2 + 14 =11.86 confidence assessment: 0
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11:03:17 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 --> I messed up by considering the x as part of the numerator. I now understand it should not have been a part of the numerator because it was not in parenthesis. After reading the explaination, I now understand the order of operations and how you came to the correct answer. self critique assessment: 2
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11:16:20 `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 --> (x - 5) ^ 2x-1 + 3 / x-2 Replace all x's with 4's (4 - 5) ^ 2(4)-1 + 3 / (4)-2 Solve for parenthesis (-1) ^ 2(4)-1 + 3 / (4) - 2 Solve for exponents 1 (4) - 1 + 3 / (4) - 2 Do multiplication and division 4 - 1 + .75 - 2 Do addition and subtraction as written =1.75 confidence assessment: 0
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11:21:18 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 --> I understand the orders of operation, although your uneven spacing in the original problem makes it confusing.
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11:22:41 *&*& 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 --> Its important to be able to read and translate both typewriter notation as well as the normal notation to know how to do the problems correctly self critique assessment: 2
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