\̮{g}X˃K?assignment #008 ⡁L{ڏy College Algebra 03-14-2006
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14:47:11 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+4)^2 )
= sqrt 4 * sqrt(x + 4)^2 = 2 * |x + 4|.................................................
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14:47:21 ** 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 | **......!!!!!!!!...................................
RESPONSE --> got it correct
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14:48:29 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
= (8 * 3) ^ 1/3 = 8^(1/3) * 3^(1/3) =2 * 3^(1/3).................................................
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14:48:40 ** (24)^(1/3) =
(8 * 3)^(1/3) = 8^(1/3) * 3^(1/3) = 2 * 3^(1/3) **......!!!!!!!!...................................
RESPONSE --> got it correct
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14:57:57 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) * (125 x^3)^(1/3) / ( 8 x^3 y^4)^(1/3)
= x^(2/3) y^(1/3) * (125x^3)^(1/3) / (8x^3 y^4)^(1/3) = x^(2/3) y^(1/3) * (5x) / (8^(1/3) x y^(4/3) = x^(2/3) y^(1/3) * (5) / (8^(1/3) y^(4/3) =5x^(2/3) / 2y^(1/4).................................................
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15:03:58 ** (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) **......!!!!!!!!...................................
RESPONSE --> I must have messed up somewhere. It looks to me that it somewhere in the beginning dropped the ^4 on the y, but I don't know how or why??? y(y^(1/3)] How did you get that when it was this...( 8 x^3y^4)^(1/3) how did the ^4 get dropped of the y or how did it cancel out. I am really confused!! Ahhh! The answer that I had was 5 x^2/3 / 2y^1/4 The only difference was the ^1/4 on the y. I must have missed a step. Oops.
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15:06:05 Extra Question: What is the simplified form of 2 sqrt(12) - 3 sqrt(27) and how did you get this result?
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RESPONSE --> 2 sqrt(12) - 3 sqrt(27)
= 2 sqrt (4*3) - 3 sqrt (3*9) = 2*2 sqrt (3) - 3*3 sqrt (3) = 4 sqrt(3) - 9 sqrt (3) = -5 sqrt (3).................................................
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15:06:27 ** 2* sqrt(12) - 3*sqrt(27) can be written as
2* sqrt (4*3) - 3 * sqrt (9*3) by factoring out the maximum possible perfect square in each square root. This simplifies to 2* sqrt (4) sqrt(3) - 3 * sqrt (9) sqrt(3) = 2*2 sqrt 3 - 3*3 * sqrt 3 = } 4*sqrt3 - 9 * sqrt3 = -5sqrt3. **......!!!!!!!!...................................
RESPONSE --> Yay I got that right. That was a lot easier than the last problem.
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15:09:16 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 --> (2 sqrt(6) + 3) ( 3 sqrt(6))
= (2 sqrt (6) * 3) * (3 * 3 sqrt (6)) = (2*3)(sqrt 6 * sqrt 6) + 9 sqrt (6) = 6 * 6 + 9 sqrt (6) = 36 + 9 sqrt (6).................................................
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15:09:26 ** (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). **......!!!!!!!!...................................
RESPONSE --> got that right too!!!
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15:12:13 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)
= [(3 / sqrt (2)] * [(sqrt (2) * sqrt (2)] = 3 sqrt (2) / (sqrt (2))^2 = 3 sqrt (2) / 2.................................................
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15:12:23 ** 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.......!!!!!!!!...................................
RESPONSE --> got it correct.
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15:29:43 Query R.8.46. What do you get when you rationalize the denominator of sqrt(3) / (sqrt(7) - sqrt(2) ) and what steps did you follow to get this result?
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RESPONSE --> sqrt (3) / (sqrt(7) - sqrt(2) )
= [sqrt 3 / (sqrt 7 - sqrt 2)] * [(sqrt 7 + sqrt 2) / (sqrt 7 + sqrt 2)] = (sqrt 3 (sqrt 7 + sqrt 2) ) / ((sqrt 7)^2 - (sqrt 2)^2) = [(sqrt 3)(sqrt 7) + (sqrt 3)(sqrt 2)] / 7 - 2 = (sqrt 21 + sqrt 6) / 5.................................................
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15:31:36 ** Starting with
sqrt(3)/(sqrt(7)-sqrt2) multiply both numerator and denominator by sqrt(7) + 2 to get (sqrt(3)* (sqrt(7) + 2))/ (sqrt(7) - 2)(sqrt(7) + 2). Since (a-b)(a+b) = a^2 - b^2 the denominator is (sqrt(7)+2 ) ( sqrt(7) - 2 ) = sqrt(7)^2 - 2^2 = 7 - 4 = 3 so we have sqrt(3) (sqrt(7) + 2) / 3.......!!!!!!!!...................................
RESPONSE --> I don't know how I messed up again but I did. Tell me where I messed up so I can fix it. Thanks!
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15:40:43 Extra Question: What steps did you follow to simplify (-8)^(-5/3) and what is your result?
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RESPONSE --> (-8)^(-5/3)
= (-8 ^(1/3)) ^-5 = -2 ^(-5) = (1/-2) ^ 5 = - 1/32.................................................
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15:40:56 ** (-8)^(-5/3) = [ (-8)^(1/3) ] ^-5. Since -8^(1/3) is -2 we get
[-2]^-5 = 1 / (-2)^5 = -1/32. **......!!!!!!!!...................................
RESPONSE --> got it correct
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15:44:39 query R.8.64. What steps did you follow to simplify (8/27)^(-2/3) and what is your result?
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RESPONSE --> (8/27)^(-2/3)
= (8 ^(-2/3)) / (27 ^(-2/3)) = (27^(2/3)) / (8 ^(2/3) = (27^(1/3))^2 / (8 ^ (1/3))^2 = 3^2 / 2^2 = 9 / 4.................................................
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15:46:29 Extra Question: What steps did you follow to simplify 6^(5/4) / 6^(1/4) and what is your result?
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RESPONSE --> got it correct.
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15:48:17 ** Use the laws of exponents (mostly x^a / x^b = x^(a-b) as follows:
6^(5/4) / 6^(1/4) = 6^(5/4 - 1/4) = 6^1 = 6. **......!!!!!!!!...................................
RESPONSE --> opps I pushed the next question/answer button twice and didn't get to answer this question.
My answer: 6^(5/4) / 6^(1/4) =6^(5/4 - 1/4) = 6.................................................
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15:52:33 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) Multiply exponents to get = x ^ (1/2)
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15:52:43 ** Express radicals as exponents and use the laws of exponents.
(x^3)^(1/6) = x^(3 * 1/6) = x^(1/2). **......!!!!!!!!...................................
RESPONSE --> got it correct
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16:02:46 Extra Question: What steps did you follow to simplify (x^(1/2) / y^2) ^ 4 * (y^(1/3) / x^(-2/3) ) ^ 3 and what is your result, assuming that x is positive and expressing your result with only positive exponents?
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RESPONSE --> (x^(1/2) / y^2) ^ 4 * (y^(1/3) / x^(-2/3) ) ^ 3
= ((x^(1/2 *4) / y^2*4) * (( y^(1/3 *3) / (x^(-2/3 *3)) = ((x^2)/ (y^8)) * ((y/x^(-2)) = x^2 * x^2 / y^8.................................................
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16:03:57 ** (x^(1/2) / y^2) ^ 4 * (y^(1/3) / x^(-2/3) ) ^ 3 =
x^(1/2 * 4) / y^(2* 4) * y^(1/3 * 3) / x^(-2/3 * 3)= x^2 / y^8 * y / x^(-2) = x^2 * x^2 / y^7 = x^4 / y^7. **......!!!!!!!!...................................
RESPONSE --> messed up near bottom. The last three parts I messed up. Can't figure it out either.
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16:11:35 query R.8.96. Factor 8 x^(1/3) - 4 x^(-2/3), x <> 0.
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RESPONSE --> 8 x^(1/3) - 4 x^(-2/3), x <> 0
= disperse (2/3) around... = (8x ^(1/3) + (2/3)) - (4x^ (-2/3) + (2/3) / x^(2/3) = 8x - 4 / x^2/3 = 4(2x - 1) / x^2/3.................................................
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16:11:54 ** To factor 8x^(1/3)- 4x^(-2/3) we first need to write the expression without negative exponents. To accomplish this we multiply through by x^(2/3) / x^(2/3), obtaining
(8 x^(1/3 + 2/3) - 4x^(-2/3 + 2/3) / x^(2/3) = (8 x - 4) / x^(2/3). We then factor 2 out of the numerator to obtain 4 ( 2x - 1) / x^(2/3). Other correct forms include: ( 4x^(1/3) ) ( 2 - ( 1/x) ) 8 x^(1/3) - 4 / x^(2/3). **......!!!!!!!!...................................
RESPONSE --> got it correct but I have no clue how!! Ha Ha
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16:13:59 Query Add comments on any surprises or insights you experienced as a result of this assignment.
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RESPONSE --> This was a relatively hard one. The 3rd question was very hard. I do need help with that. I went through the CD's that came with the book, but I still couldn't figure it out. A couple others I must be missing some steps or something. Thanks for your patience with me.