course Mth 158 ]jvcӍň`qassignment #008
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17:03:46 **** query R.8.12. Simplify the cube root of 54
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RESPONSE --> The square root of 54= the square root of 6*9 = 6*the square root of 3.
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17:07:54 The cube root of 54 is expressed as 54^(1/3). The number 54 factors into 2 * 3 * 3 * 3, i.e., 2 * 3^3. Thus 54^(1/3) = (2 * 3^3) ^(1/3) = 2^(1/3) * (3^3)^(1/3) = 2^(1/3) * 3^(3 * 1/3) = 2^(1/3) * 3^1 = 3 * 2^(1/3), i.e., 3 * cube root of 2.
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RESPONSE --> I'm trying to understand this.
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17:16:16 **** query R.8.18. Simplify the cube root of (3 x y^2 / (81 x^4 y^2) ).
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RESPONSE --> The cube root of 3xy^2 /81 x^4y =3(4) /3 x^4
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17:36:13 Extra Question: What is the simplified form of (2 sqrt(6) + 3) ( 3 sqrt(6)) and how did you get this result?
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RESPONSE --> The simplified form of 2sqrt (6) + 3 would be the cube of 9 which is 3 and the 3 squared * 6 would be the square of 18 = the square root of 9 x 3. The answer being 3 times the square root of 3.Enter, as appropriate, an answer to the question, a critique of your answer in response to a given answer, your insights regarding the situation at this point, notes to yourself, or just an OK. Always critique your solutions by describing any insights you had or errors you makde, and by explaining how you can make use of the insight or how you now know how to avoid certain errors. Also pose for the instructor any question or questions that you have related to the problem or series of problems.
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17:38:10 ** (2*sqrt(6) +3)(3*sqrt(6)) expands by the Distributive Law to give (2*sqrt(6) * 3sqrt(6) + 3*3sqrt(6)), which we rewrite as (2*3)(sqrt6*sqrt6) + 9 sqrt(6) = (6*6) + 9sqrt(6) = 36 +9sqrt(6). **
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RESPONSE --> ok
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17:42:31 **** query R.8. Expand (sqrt(x) + sqrt(5) )^2
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RESPONSE --> The square of x is x and the square of root of 5^2 is 25 =5 so the answere is x=5.
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17:44:13 (sqrt(x) + sqrt(5) )^2 = (sqrt(x) + sqrt(5) ) * (sqrt(x) + sqrt(5) ) = sqrt(x) * (sqrt(x) + sqrt(5) ) + sqrt(5) * (sqrt(x) + sqrt(5) ) = sqrt(x) * sqrt(x) + sqrt(x) * sqrt(5) + sqrt(5) * sqrt(x) + sqrt(5) * sqrt(5) = x + 2 sqrt(x) sqrt(5) + 5.
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RESPONSE --> Ok
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17:48:42 Query R.8.42. What do you get when you rationalize the denominator of 3 / sqrt(2) and what steps did you follow to get this result?
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RESPONSE --> 3/sqrt (2) is equal 3/ sqrt of 2 times the sqrt of 2/ sqrt of 2 = sqrt of 9 / 2
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17:49:48 ** Starting with 3/sqrt(2) we multiply numerator and denominator by sqrt(2) to get (2*sqrt(2))/(sqrt(2)*sqrt(2)) = (3 sqrt(2) ) /2.
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RESPONSE --> ok
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17:57:31 **** query R.8.48. Rationalize denominator of sqrt(2) / (sqrt(7) + 2)
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RESPONSE --> The sqrt of 2/sqrt (7) +2 times sqrt of 2 * sqrt of (7) +2 =sqrt 2(sqrt(7) -2)/ 49-2= sqrt (2) (sqrt7 -2/47)
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17:58:40 To rationalize the denominator sqrt(7) + 2 we multiply both numerator and denominator by sqrt(7) - 2. We obtain ( sqrt(2) / (sqrt(7) + 2) ) * (sqrt(7) - 2) / (sqrt(7) - 2) = sqrt(2) * (sqrt(7) - 2) / ( (sqrt(7) + 2) * ( sqrt(7) - 2) ) = sqrt(2) * (sqrt(7) - 2) / (sqrt(7) * sqrt(7) - 4) = sqrt(2) * (sqrt(7) - 2 ) / (7 - 4) = sqrt(2) * (sqrt(7) - 2 ) / 3.
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RESPONSE --> ok
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18:12:09 Extra Question: What steps did you follow to simplify (x^3)^(1/6) and what is your result, assuming that x is positive and expressing your result with only positive exponents?
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RESPONSE --> (x^3)^(1/6) = (x^3)^1/6 = x^3/6 = X^1/2 = sqrt x
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18:12:57 ** Express radicals as exponents and use the laws of exponents. (x^3)^(1/6) = x^(3 * 1/6) = x^(1/2). **
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RESPONSE --> ok
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18:15:47 **** query R.8.60. Simplify 25^(3/2).
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RESPONSE --> 25^3/2 = (sqrt 25) ^ 3 = 5^ 3 =125
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18:15:57 25^(3/2) = (5^2)^(3/2) = 5^(2 * 3/2) = 5^(2 * 3/2) = 5^3.
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RESPONSE --> ok
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18:26:39 **** query R.8.72. Simplify and express with only positive exponents: (xy)^(1/4) (x^2 y^2) ^(1/2) / (x^2 y)^(3/4).
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RESPONSE --> (xy) 1/4 (x^2y^2)/(x^2y)^3/4 = (x^1/4)(y^1/4) (x^2)^1/2(y^2)^1/2 /(x^2) ^3/4 (y)^3/4= (x)^ 1/4 (y)^1/4 (x) (y)/ (x^)^3/4 (y) ^3/4= (x) (y) (x)^1/4 (Y) ^1/4
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18:26:51 (xy)^(1/4) (x^2 y^2) ^(1/2) / (x^2 y)^(3/4) = x^(1/4) * y^(1/4) * (x^2)^(1/2) * y^2 ^ (1/2) / ( (x^2)^(3/4) * y^(3/4) ) = x^(1/4) * y^(1/4) * x^(2 * 1/2) * y^(2 * 1/2) / ( (x^(2 * 3/4) * y^(3/4) ) = x^(1/4) y^(1/4) * x^1 * y^1 / (x^(3/2) y^(3/4) ) = x^(1 + 1/4) y^(1 + 1/4) / (x^(3/2) y^(3/4) ) = x^(5/4) y^(5/4) / (x^(3/2) y^(3/4) ) = x^(5/4 - 3/2) y^(5/4 - 3/4) = x^(5/4 - 6/4) y^(2/4) = x^(-1/4) y^(1/2) = y^(1/2) / x^(1/4).
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RESPONSE --> ok
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18:46:25 **** query R.8.84. Express with positive exponents: ( (9 - x^2) ^(1/2) + x^2 ( 9 - x^2) ^(-1/2) ) / (9 - x^2).
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RESPONSE --> =(9-x^2)^1/2 = X^2/(9-x)^1/2/ (9-x) 1/2= (9-x)^1/2 (9-x)1/2 + x^2/ 9-x^1/2/ 9-x^2
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18:46:39 ( (9 - x^2) ^(1/2) + x^2 ( 9 - x^2) ^(-1/2) ) / (9 - x^2) =
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RESPONSE --> ok
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18:49:04 **** query R.8.108. v = sqrt(64 h + v0^2); find v for init vel 0 height 4 ft; for init vel 0 and ht 16 ft; for init vel 4 ft / s and height 2 ft.
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RESPONSE --> I'm lost
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18:49:18 If initial velocity is 0 and height is 4 ft then we substitute v0 = 0 and h = 4 to obtain v = sqrt(64 * 4 + 0^2) = sqrt(256) =16.+vbcrlf+vbcrlf+If initial velocity is 0 and height is 16 ft then we substitute v0 = 0 and h = 4 to obtain v = sqrt(64 * 16 + 0^2) = sqrt(1024) = 32. Note that 4 times the height results in only double the velocity.+vbcrlf+vbcrlf+If initial velocity is 4 ft / s and height is 2 ft then we substitute v0 = 4 and h = 2 to obtain v = sqrt(64 * 2 + 4^2) = sqrt(144) =12.
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RESPONSE --> ok
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18:52:31 Extra Question: What is the simplified form of (24)^(1/3) and how did you get this result?
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RESPONSE --> (24)^ (1/3) = (cube root of 24) =
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18:53:33 ** (24)^(1/3) = (8 * 3)^(1/3) = 8^(1/3) * 3^(1/3) = 2 * 3^(1/3) **
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RESPONSE --> ok
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19:03:52 Extra Question: What is the simplified form of (x^2 y)^(1/3) * (125 x^3)^(1/3) / ( 8 x^3 y^4)^(1/3) and how did you get this result?
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RESPONSE --> (x^2 y)^1/3 * (125x^3)^1/3/ (8x^3y^4) ^1/3 = X^2/3 y^1/3 * sqrt 125 x/8xy^4/3= 5^3x^2/3 y^1/3/2^3y^4/3
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19:04:02 ** (x^2y)^(1/3) * (125x^3)^(1/3)/ ( 8 x^3y^4)^(1/3) (x^(2/3)y^(1/3)* (5x)/[ 8^(1/3) * xy(y^(1/3)] (x^(2/3)(5x) / ( 2 xy) 5( x^(5/3)) / ( 2 xy) 5x(x^(2/3)) / ( 2 xy) 5 ( x^(2/3) ) / (2 y) **
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RESPONSE --> ok
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19:09:52 Extra question. What is the simplified form of sqrt( 4 ( x+4)^2 ) and how did you get this result?
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RESPONSE --> =sqrt 4 (x^2) (4^2)= (sqrt 4) (x^2) (sqrt4)(4^2)=2(sqrt 4)(x^2)
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19:10:17 ** sqrt(a b) = sqrt(a) * sqrt(b) and sqrt(x^2) = | x | (e.g., sqrt( 5^2 ) = sqrt(25) = 5; sqrt( (-5)^2 ) = sqrt(25) = 5. In the former case x = 5 so the result is x but in the latter x = -5 and the result is | x | ). Using these ideas we get sqrt( 4 ( x+4)^2 ) = sqrt(4) * sqrt( (x+4)^2 ) = 2 * | x+4 | **
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RESPONSE --> ok
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19:11:54 Query Add comments on any surprises or insights you experienced as a result of this assignment.
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RESPONSE --> The questions from the query were not on the list of homework questions so I had to work each questions out. this has been a very time consuming assignment and I know I haven't comprehended it like I would have wanted to.
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