Assignment 8

course Mth 158

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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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19:21:09

** 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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19:26:52

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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19:26:56

** (24)^(1/3) =

(8 * 3)^(1/3) =

8^(1/3) * 3^(1/3) =

2 * 3^(1/3) **

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19:51:29

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)^(1/3) (y)^(1/3) * 5x / (8)^(1/3) * xy(y^(1/3))

x^(2/3)*y^(1/3) * 5x / 2xy*y^(1/3)

x^(2/3)* 5x / 2xy

x^(2/3)* 5 / 2y

x^(2/3)* 5 / (2y).

x^(2/3)* 5 / 2y means ( x^(2/3)* 5 / 2 ) * y

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19:51:53

** (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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19:59:01

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) * sqrt(3) - 3 * sqrt(9) * sqrt(3)

2 * 2 *sqrt(3) - 3 * 3 * sqrt(3)

4 * sqrt(3) - 9 * sqrt(3)

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19:59:13

** 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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20:12:21

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 sqrt(6) + 3(3 sqrt(6))

2 * 3 * (sqrt6)^2 + 9 sqrt(6)

6 * 6 + 9 sqrt(6)

36 + 9 sqrt(6)

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20:12:35

** (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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20:16:01

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) / 2

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20:16:15

** 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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20:24:12

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) + 2 / sqrt(7)-2 * sqrt(7)+2

sqrt(3) * sqrt(7) + 2 / 7 - 4

sqrt(3) * sqrt(7) + 2 / 3

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20:24:17

** 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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20:30:15

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)

-2^ -5

1/-2^5 = -1/32

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20:30:22

** (-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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20:33:44

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^(2/3) / 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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20:34:17

** 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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20:36:36

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^(5/4 - 1/4)

6^(4/4) = 6^1 = 6

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20:36:40

** 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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20:39:05

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)

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20:39:13

** 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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20:53:33

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^(3/3) / x^(-6/3) )

(x^(2) / y^(8)) * (y / x^(-2))

x^2 * y * x^2 / y^8

x^4 / y^7

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20:53:37

** (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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20:54:35

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) =

8 x^(1/3) - 4 / x^(2/3)

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20:54:42

** 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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20:55:05

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

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none

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You did very well on these questions. I'll continue to nag you occasionally about grouping, but it's clear that you understand everything so far.

Let me know if you have questions.

Assignment 8

x^(2/3)* 5 / (2y). x^(2/3)* 5 / 2y means ( x^(2/3)* 5 / 2 ) * y

You did very well on these questions. I'll continue to nag you occasionally about grouping, but it's clear that you understand everything so far. Let me know if you have questions.

x^(2/3)* 5 / (2y).

x^(2/3)* 5 / 2y means ( x^(2/3)* 5 / 2 ) * y

You did very well on these questions. I'll continue to nag you occasionally about grouping, but it's clear that you understand everything so far.

Let me know if you have questions.