course Mth 158 assignment #008 008. `query 8 College Algebra 02-06-2007
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13:38:37 R.8.64. What is the simplified form of sqrt( 4 ( x+4)^2 ) and how did you get this result?
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RESPONSE --> confidence assessment: 0
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13:43:47 ** (24)^(1/3) = (8 * 3)^(1/3) = 8^(1/3) * 3^(1/3) = 2 * 3^(1/3) **
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RESPONSE --> ok self critique assessment: 2.
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13:50:50 ** (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 self critique assessment: 2
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13:51:45 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 --> ok confidence assessment: 0
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13:53:17 ** 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. **
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RESPONSE --> ok self critique assessment: 2
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13:54:05 R.8.78. 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 --> ok confidence assessment: 0
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13:56:28 ** (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 self critique assessment: 2
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FȂKT{mw assignment #008 008. `query 8 College Algebra 02-06-2007
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14:05:48 R.8.64. 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| confidence assessment: 2
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14:06:09 ** 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 --> Okay! I finally understand. self critique assessment: 2
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14:10:37 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) confidence assessment: 2
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14:10:44 ** (24)^(1/3) = (8 * 3)^(1/3) = 8^(1/3) * 3^(1/3) = 2 * 3^(1/3) **
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RESPONSE --> self critique assessment: 2
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14:25:35 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^2y)^(1/3) * (125x^3)^(1/3) / (8x^3y^4)(1/3) = = (x^2)^(1/3) * y^(1/3) * (125)^(1/3)(x^3)^(1/3) / 8^(1/3) * (x^3)^(1/3) * (y^3)^(1/3) * y^(1/3) =x^(2/3) * y^(1/3) * 5x / 2xy * y^(1/3) =5x^(3/3) * x^(2/3) / 2xy =5x^(2/3) / 2y confidence assessment: 2
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14:25:53 ** (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 self critique assessment: 2
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14:31:26 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 --> 2sqrt(12) - 3sqrt(27)= Find like radicals for combinding 2sqrt(4*3) - 3sqrt(9*7) or 2sqrt(4) * sqrt(3) - 3sqrt(9) * sqrt(3)= 2 * 2 * sqrt(3) - 3 * 3 * sqrt(3)= 4sqrt(3) - 9sqrt(3)= -5sqrt(3) confidence assessment: 2
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14:31:40 ** 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. **
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RESPONSE --> Yes! self critique assessment: 2
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14:38:04 ** (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 --> Okay, to be understood by using the distributive law. (3 * sqrt(6)) to be multiplied by both (2 * sqrt(6)) and 3 (3 * sqrt(6)) * (2 * sqrt(6)) + (3 * sqrt(6)) * (3) = (3*2)(sqrt(6) * sqrt(6)) + (3*3) * sqrt(6) = (6 * 6) + 9sqrt(6)= 36 + 9sqrt(6) self critique assessment: 2
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14:42:07 R.8.90. What do you get when you rationalize the denominator of 5 / sqrt(10) and what steps did you follow to get this result?
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RESPONSE --> To rationalize the denominator of 5/sqrt(10), it is that both the denominator and numerator to be multiplied by sqrt(10). 5/sqrt(10) * sqrt(10)/sqrt(10)= 5sqrt(10)/(sqrt(10))^2= 5sqrt(10)/10+ sqrt(10)/2 confidence assessment: 2
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14:42:40 ** Starting with 5/sqrt10 we multiply numerator and denominator by sqrt(10) to get (5*sqrt10)/(sqrt10*sqrt10) = (5sqrt10)/10 = sqrt10/2 **
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RESPONSE --> whoooah! ok I am really getting this... self critique assessment: 2
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14:48:23 R.8.96. What do you get when you rationalize the denominator of 10 / (4 - sqrt(2) ) and what steps did you follow to get this result?
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RESPONSE --> To rationalize the denominator of 10/(4-sqrt(2)), it is to be multiplied by both the numerator and denominator by (4+sqrt(2)). 10(4+sqrt(2)) / (4)^2 - (sqrt2)^2= 40+10sqrt(2) / 16-2= 40+10sqrt(2) / 14 confidence assessment: 2
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14:49:46 ** Starting with 10/(4-sqrt2) multiply both numerator and denominator by 4 + sqrt(2) to get (10* (4+sqrt2))/ (4-sqrt2)(4+sqrt2). Since (a-b)(a+b) = a^2 - b^2 the denominator is (4+sqrt(2) ) ( 4 - sqrt(2) ) = 16 - 2 = 14 so we have (40+ 10sqrt2) / 14. Dividing numerator and denominator by 2 we end up with (20 + 5 sqrt(2) / 7 **
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RESPONSE --> Okay so it was to be simplified once more by 2 giving (20 + 5sqrt(2)) / 7 self critique assessment: 2
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14:50:45 Add comments on any surprises or insights you experienced as a result of this assignment.
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RESPONSE --> At first I had a hard time understanding, but finally got the biggests part of what I was missing. confidence assessment: 3
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16:35:57 R.9.6. What steps did you follow to simplify (-8)^(-5/3) and what is your result?
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RESPONSE --> a^(m/n)=nsqrt(a)^m=(nsqrt(a))^m (-8)^(-5/3)= (3sqrt(-8))^-5= (-2)^-5= -1/32 confidence assessment: 2
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16:36:32 ** (-8)^(-5/3) = [ (-8)^(1/3) ] ^-5. Since -8^(1/3) is -2 we get [-2]^-5 = 1 / (-2)^5 = -1/32. **
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RESPONSE --> yes! Finally I'm getting it. self critique assessment: 2
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16:45:13 R.9.12. What steps did you follow to simplify (8/27)^(-2/3) and what is your result?
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RESPONSE --> ok confidence assessment: 0
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16:48:15 ** Starting with (8/27)^(-2/3) we can write as (8^(-2/3)/27^(-2/3)). Writing with positive exponents this becomes (27^(2/3)/8^(2/3)) 27^(2/3) = [ 27^(1/3) ] ^2 = 3^2 = 9 and 8^(2/3) = [ 8^(1/3) ] ^2 = 2^2 = 4 so the result is (27^(2/3)/8^(2/3)) = 9/4. **
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RESPONSE --> That was a little confusing. self critique assessment: 2
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16:53:47 R.9.24. What steps did you follow to simplify 6^(5/4) / 6^(1/4) and what is your result?
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RESPONSE --> a^m / a^n = (a)^m-n Law of Exponents 6^(5/4) / 6^(1/4) = 6^(5/4 - 1/4)= 6 confidence assessment: 2
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16:54:13 ** 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. **
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RESPONSE --> Yeah!..... self critique assessment: 3
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17:25:59 R.9.36. 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 --> Law of exponents (x^3)^(1/6)= x^(1/2) confidence assessment: 2
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17:26:12 ** 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 --> yes... self critique assessment: 2
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17:38:11 R.9.48. 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 --> law of exponents (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/y^8 * x^2/y= x^(2+2)/y^(8+1)= x^4/y^9 confidence assessment: 3
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17:39:40 ** (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. **
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RESPONSE --> Ok I added the exponents of y 8+1=9 instead of subtracting 8-1=7 self critique assessment: 2
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17:44:42 R.9.72. Factor 8 x^(1/3) - 4 x^(-2/3), x <> 0.
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RESPONSE --> OK confidence assessment: 1
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17:48:51 ** 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). **
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RESPONSE --> A little confusing. self critique assessment: 2
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