Assignment 8

Good work, though there were a few discrepancies between your solution and the given solutions that should have been noted and self-critiqued. I understand that you're under pressure and you clearly understand this pretty well, so for now I'll leave it to you to judge when it's necessary to self-critique.

Let me know if anything is unclear, and include specifics about what you do and do not understand.

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19:06:04 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 ) 4 is a perfect square = 2 (sqrt (x+4)^2) (x+4) is also a square = 2 (x+4)

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19:06:13 ** 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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19:09:31 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) using a^(m/n) = 24^(1/3) = 8^(1/3) * 3^(1/3) factored for cubes = 2 * 3^(1/3)

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19:09:40 ** (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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19:24:53 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) = (125 x^3)^1/3 / (8 x y^3)^1/3 = [(5 x) / (2 y (sqrt x))] * (sqrt x) / (sqrt x) = [5 (x^(3/2))] / (2 xy)

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19:26:31 ** (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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20:52:51 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))] factoring for squares = [2 * 2 * sqrt 3] - [3 * 3 * sqrt 3] = 4 (sqrt 3) - 9 (sqrt 3) = -5 (sqrt 3)

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20:53:02 ** 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

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20:57:11 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)) = 6 (sqrt 6)^2 + 9 (sqrt 6) = 36 + 9 (sqrt 6)

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20:57:19 ** (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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21:00:04 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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21:00:15 ** 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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21:08:21 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 21) + (sqrt 6)] / [(sqrt 7)^2 - (sqrt 2)^2] = (sqrt 27) / 5 = (sqrt (9 * 3) / 5 = [3 (sqrt 3)] / 5

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21:10:12 ** 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.

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RESPONSE --> ok

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21:19:39 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) = (cubert -8)^-5 = -2^-5 = 1 / (-2^5) = - 1 / 32

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21:19:48 ** (-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

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21:24:51 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) = (27/8)^(2/3) = [(3^3)^(2/3)] / [(2^3)^(2/3)] = [(3^(6/3)] / [(2^(6/3)] = (3^2) / (2^2) = 9 / 4

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21:24:59 ** 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

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21:51:58 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) = 6^(4/4) = 6

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21:52:11 ** 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

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21:58:41 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^(3/6) = x^(1/2)

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21:58:48 ** 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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08:19:13 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) / y^2) ^ 4 * [(y^(1/3)) / [(1 / x)^(2/3))]^3] = (x^(1/2) / y^2) ^ 4 * [(y^(1/3)) / 1 * (x^(2/3)) / 1]^3 = (x^2) / (y^8) * yx^2 = (2x^2y) / (y^8) = (2x^2) / (y^7)

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08:22:38 ** (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

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08:27:37 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 = 8 x^(1/3) - [4 / (x^(2/3))] = [8 x^(1/3) * x^(2/3) - 4] / (x^(2/3)) = (8x - 4) / (x^(2/3) = [4 (2x - 1)] / (x^(2/3))

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08:28:11 ** 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 --> ok

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08:28:40 Query Add comments on any surprises or insights you experienced as a result of this assignment.

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RESPONSE --> none

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