course MTH 158 哆ÄÎÐùZÉîÖÔá«ùĉ…ç|„Þassignment #008
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12:19:57 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 --> The sqrt of 4 is 2, and the sqrt of (x+4)^2 is x+4, so... 2 * | x+4 | confidence assessment: 2
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12:20:04 ** 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 self critique assessment: 3
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12:20:30 Extra Question: What is the simplified form of (24)^(1/3) and how did you get this result?
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RESPONSE --> The rule for this is a^m/n = ^n sqrt a^m. So... (24)^(1/3) = cubed root (24) cubed root (24) = cubed root (8 * 3), which gives us 2 cubrt 3 confidence assessment: 2
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12:20:37 ** (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: 3
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12:23:41 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 * qubrt y) * 5x / 2xy cubrt y which cancels out to x^2/3 * 5 / 2y confidence assessment: 2
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12:23:58 ** (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) **
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RESPONSE --> ok self critique assessment: 3
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12:27:28 R.8.28:. 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 * 2 sqrt(3) - 3 * 3 sqrt(3) = 4 sqrt(3) - 9 sqrt(3) = -5 sqrt(3) confidence assessment: 2
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12:27:41 ** 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: 3
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12:44:39 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 --> (3 sqrt(6)) * (2 sqrt(6) + 3) = 36 + 9 sqrt(6) confidence assessment: 2
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12:44:53 ** (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: 3
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12:55:09 Extra Qustion. 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 --> Multiplying the num and denom by sqrt(10) we get 5 sqrt(10) / 10, which goes down to sqrt(10) / 2 confidence assessment: 2
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12:55:18 ** 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 --> ok self critique assessment: 3
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13:00:16 Extra Question. 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 --> Multiply the num and denom by (4 + sqrt(2) ) 10 (4 + sqrt(2) ) / 16 - 4sqrt(2) + 4sqrt(2) + 2 = 10 (4 + sqrt(2) ) / 18 confidence assessment: 2
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13:03:41 ** 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 --> understood self critique assessment: 3
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22:36:13 Extra Question: What steps did you follow to simplify (-8)^(-5/3) and what is your result?
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RESPONSE --> The rule for this is a^m/n = ^n sqrt a^m. So... (-8)^(-5/3) = cubrt (5/8) confidence assessment: 1
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22:36:58 ** (-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 --> ok self critique assessment: 1
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11:12:30 R.8.62. What steps did you follow to simplify (8/27)^(-2/3) and what is your result?
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RESPONSE --> Turn 8/27 into its reciprocal (8/27)^(-2/3) = (27/8)^(2/3) Apply the exponent cubrt (27)^2 / cubrt (8)^2 = cubrt (729) / cubrt (64) = 9/4 confidence assessment: 1
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11:12:51 ** 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 --> ok self critique assessment: 3
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11:19:19 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 --> 6^(5/4) / 6^(1/4) = ^4 root (6)^5 / ^4 root (6) ^4 root of (6)^5 = 6 ^4 root (6), so... 6 ^4 root (6) / ^4 (6) = 6 confidence assessment: 1
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11:19:32 ** 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 --> ok self critique assessment: 3
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11:22:46 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 --> (x^3)^(1/6) = x^3/6 x^3/6 = x^1/2 x^1/2 = sqrt (x) confidence assessment: 1
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11:22:58 ** 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 self critique assessment: 3
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11:27:26 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^2 / y^8 * x^-6/3 = x^2 / y^8 * x^-2 = x^2 / y^8 * 2/x = 2x / y^8 confidence assessment: 1
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11:30:20 ** (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 --> Understood - I misread the question and figured it out using (x^(1/2) / y^2) ^ 4 * x^(-2/3) ) ^ 3 rather than (x^(1/2) / y^2) ^ 4 * (y^(1/3) / x^(-2/3) ) ^ 3 self critique assessment: 2
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12:12:43 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) = cubrt(8x) - cubrt(4x^-2) = 2 cubrt(x) - cubrt [(4)(2/x)] confidence assessment: 1
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12:18:48 ** 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 --> understood self critique assessment: 2
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12:18:55 Add comments on any surprises or insights you experienced as a result of this assignment.
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RESPONSE --> confidence assessment: 1
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