Assignment 8

course mth 158

q????j{{?L??X????assignment #008008. `query 8

College Algebra

07-04-2008

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02:50:01

**** query R.8.12. Simplify the cube root of 54

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RESPONSE -->

cube root of 54

cube root of 27 *2

=3*cube root of 2

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02:50:15

The cube root of 54 is expressed as 54^(1/3).

The number 54 factors into 2 * 3 * 3 * 3, i.e., 2 * 3^3. Thus

54^(1/3) = (2 * 3^3) ^(1/3)

= 2^(1/3) * (3^3)^(1/3)

= 2^(1/3) * 3^(3 * 1/3)

= 2^(1/3) * 3^1

= 3 * 2^(1/3), i.e., 3 * cube root of 2.

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02:53:18

**** query R.8.18. Simplify the cube root of (3 x y^2 / (81 x^4 y^2) ).

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RESPONSE -->

(3 x y^2) / (81 x^4 y^2) ).

(3 x y^2) / (27*3*x^3*x*y^2)

(3 x y^2) / (3 x y^2) (27x)

=27x

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02:56:24

The cube root of (3 x y^2 / (81 x^4 y^2) ) is

(3 x y^2 / (81 x^4 y^2) ) ^ (1/3) = (1 / (27 x^3) ) ^(1/3)

= 1 / ( (27)^(1/3) * ^x^3^(1/3) )

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

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

= 1 / (3 * x)

= 1 / (3x).

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RESPONSE -->

i see i forgot to put the cube root into the problem.

so the cube root of 1/27x^3 should be 1/(3x)

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02:59:09

**** query R.8.30. Simplify 2 sqrt(12) - 3 sqrt(27).

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RESPONSE -->

2 sqrt of 12 - 3 sqrt of 27

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

4 sqrt of (3) -9 sqrt of (3)

=-5 sqrt of 3

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02:59:26

2 sqrt(12) - 3 sqrt(27) = 2 sqrt( 2*2*3) - 3 sqrt(3*3*3)

= 2 sqrt(2^2 * 3) - 3 sqrt(3^3)

= 2 sqrt(2^2) sqrt^3) - 3 sqrt(3^2) sqrt(3)

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

= 4 - 9 sqrt(3).

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03:06:54

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 -->

you mulitply the 3 sqrt of 6 to the 2 sqrt of (6) +3 and get 3 sqrt of 6 (2 sqrt of 6) +3(3 sqrt of 6)

6sqrt of 6^2+9 sqrt of 6

36 + 9 sqrt of 6

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03:06:59

** (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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03:08:28

**** query R.8. Expand (sqrt(x) + sqrt(5) )^2

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RESPONSE -->

(sqrt(x) + sqrt(5) )^2

sqrt x^2 + sqrt 5^2

x+5

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03:10:32

(sqrt(x) + sqrt(5) )^2 = (sqrt(x) + sqrt(5) ) * (sqrt(x) + sqrt(5) )

= sqrt(x) * (sqrt(x) + sqrt(5) ) + sqrt(5) * (sqrt(x) + sqrt(5) )

= sqrt(x) * sqrt(x) + sqrt(x) * sqrt(5) + sqrt(5) * sqrt(x) + sqrt(5) * sqrt(5)

= x + 2 sqrt(x) sqrt(5) + 5.

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RESPONSE -->

ok i see what i did wrong.

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03:11:55

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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03:11:58

** 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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03:14:25

**** query R.8.48. Rationalize denominator of sqrt(2) / (sqrt(7) + 2)

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RESPONSE -->

sqrt(2) / (sqrt(7) + 2)

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

sqrt(2)(sqrt(7) - 2) /7-4

sqrt(2)(sqrt(7) - 2) /3

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03:14:30

To rationalize the denominator sqrt(7) + 2 we multiply both numerator and denominator by sqrt(7) - 2. We obtain

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

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

= sqrt(2) * (sqrt(7) - 2) / (sqrt(7) * sqrt(7) - 4)

= sqrt(2) * (sqrt(7) - 2 ) / (7 - 4)

= sqrt(2) * (sqrt(7) - 2 ) / 3.

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RESPONSE -->

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03:18:13

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 -->

3/1*6/6=18/6) and 18 /6=3 so 3/6=12

so x^3)^(1/6) =x^(1/2)

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03:18:19

** 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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03:18:59

**** query R.8.60. Simplify 25^(3/2).

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RESPONSE -->

25^(3/2).

sqrt of 25 is 5

and 5^3 is 125

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03:19:15

25^(3/2) = (5^2)^(3/2) = 5^(2 * 3/2) = 5^(2 * 3/2) = 5^3.

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03:26:05

**** query R.8.72. Simplify and express with only positive exponents: (xy)^(1/4) (x^2 y^2) ^(1/2) / (x^2 y)^(3/4).

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RESPONSE -->

xy)^(1/4) (x^2 y^2) ^(1/2) / (x^2 y)^(3/4).

x^(1/4)y^(1/4)](xy)/x^(6/4)y^(3/4)

x^(1/4)y^(1/4)](xy)/x^(1/4)y^(1/4)](x^(5/4y^(1/2)

(xy)/(x^(5/4y^(1/2)

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03:28:03

(xy)^(1/4) (x^2 y^2) ^(1/2) / (x^2 y)^(3/4)

= x^(1/4) * y^(1/4) * (x^2)^(1/2) * y^2 ^ (1/2) / ( (x^2)^(3/4) * y^(3/4) )

= x^(1/4) * y^(1/4) * x^(2 * 1/2) * y^(2 * 1/2) / ( (x^(2 * 3/4) * y^(3/4) )

= x^(1/4) y^(1/4) * x^1 * y^1 / (x^(3/2) y^(3/4) )

= x^(1 + 1/4) y^(1 + 1/4) / (x^(3/2) y^(3/4) )

= x^(5/4) y^(5/4) / (x^(3/2) y^(3/4) )

= x^(5/4 - 3/2) y^(5/4 - 3/4)

= x^(5/4 - 6/4) y^(2/4)

= x^(-1/4) y^(1/2)

= y^(1/2) / x^(1/4).

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RESPONSE -->

ok i see what i did wrong i should of kept the ^1 and i omited it.

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03:59:27

**** query R.8.84. Express with positive exponents: ( (9 - x^2) ^(1/2) + x^2 ( 9 - x^2) ^(-1/2) ) / (9 - x^2).

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RESPONSE -->

???

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04:00:05

( (9 - x^2) ^(1/2) + x^2 ( 9 - x^2) ^(-1/2) ) / (9 - x^2)

=

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RESPONSE -->

I can't figure this one out

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04:08:49

**** query R.8.108. v = sqrt(64 h + v0^2); find v for init vel 0 height 4 ft; for init vel 0 and ht 16 ft; for init vel 4 ft / s and height 2 ft.

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RESPONSE -->

v = sqrt(64 h + v0^2

v = sqrt(64 (4) + v0^2

v= sqrt 256

v=16

v = sqrt(64 (16) + v0^2

v= sqrt 1024

v= 32

v = sqrt(64 (2) + 4

v = sqrt(128+4)

v = sqrt(1320

v= 11.489

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04:10:06

If initial velocity is 0 and height is 4 ft then we substitute v0 = 0 and h = 4 to obtain

v = sqrt(64 * 4 + 0^2) = sqrt(256) =16.+vbcrlf+vbcrlf+If initial velocity is 0 and height is 16 ft then we substitute v0 = 0 and h = 4 to obtain

v = sqrt(64 * 16 + 0^2) = sqrt(1024) = 32.

Note that 4 times the height results in only double the velocity.+vbcrlf+vbcrlf+If initial velocity is 4 ft / s and height is 2 ft then we substitute v0 = 4 and h = 2 to obtain

v = sqrt(64 * 2 + 4^2) = sqrt(144) =12.

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RESPONSE -->

ok i didnot square the velocity in the last problem

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04:11:15

Extra Question: What is the simplified form of (24)^(1/3) and how did you get this result?

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RESPONSE -->

cube root of 24

cube root of 8 * 3

2*cube root of 3

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04:11:33

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

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

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

2 * 3^(1/3) **

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04:29:54

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)y^)*125x^3)/8xy^4)^1/3

[(125x^5y)/(8xy^4)]^1/3

9125x^(5/3)y^(1/3)/(8x^(1/3)y^(4/3))

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04:42:00

** (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 i see how you work the problem

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04:43:56

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(x^2+4^2)

2(x+4)

2x+8

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04:44:03

** 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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04:44:14

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

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Your work looks very good. Let me know if you have any questions. &#