Review assignment 8

course Mth 158

It took me two days to complete this assignment. I haven't had any college math since 1978 when I graduated from CVC (UVA/wise). I work on this class daily, but I'm concerned that the pace at which I am completing the assignments is not keeping pace with the expectations of the timetable. My teaching obligations were not fullfiled until June 6. I am willing to work hard, however, I don't think I can go any faster. Do you think I should continue it or try to pick it up this fall?

∓ݐ{rzwzŧ΍assignment #008

Your work has been received. Please scroll through the document to see any inserted notes (inserted at the appropriate place in the document, in boldface) and a note at the end. The note at the end of the file will confirm that the file has been reviewed; be sure to read that note. If there is no note at the end, notify the instructor through the Submit Work form, and include the date of the posting to your access page.

008. `query 8

College Algebra

06-18-2008

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19:23:43

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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RESPONSE -->

I thought the cube root of 54 could be written as the cube root of 27* the cube root of two.

It can, but the cube root of 27 is 3.

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19:26:34

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

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RESPONSE -->

1/ 27x^3

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ϡᝳȣͧye

assignment #008

008. `query 8

College Algebra

06-18-2008

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19:36:02

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

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RESPONSE -->

the cube root of 54 = cube root of 27 * cube root of 2

or, 3 * cub root of 2.

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19:36:43

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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RESPONSE -->

ok

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19:51:20

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

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RESPONSE -->

1/ (3 *cube root of 30

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20:06:07

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 -->

obviously I' misssing something.

I stated 3^ 1/3 * x^1/3 *(y^)^1/3 / [ 3^1/3* 27^1/3 * y^1/3. I then cancelled the common parts of the numerator and denominator. I don't see my mistake.

3^ 1/3 * x^1/3 *(y^)^1/3 / [ 3^1/3* 27^1/3 * y^1/3] would be the cube root of

(3 x y / (81 y) ).

Assuming the problem is as stated above, you're missing the cube root of the x^4 in the denominator, and in both cases your y^(1/3) should be (y^2)^(1/3) or y^(2/3).

You have most of the right ideas; be careful about the details and let me know if there's anything you don't understand.

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20:21:05

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

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RESPONSE -->

2* sqrt 4 * sqrt 3 -( 3 * sqrt 9 *sqrt3)

4* sqrt 3 -( 9 * sqrt 3)

-5sqrt 3

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22:15:09

** (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 -->

Not sure where this solution came from, but your solution is correct.

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22:23:31

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

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RESPONSE -->

( sqrtx )^2 *( 5^2)

x +25

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22:42:00

06-18-2008 22:42:00

(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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NOTES ------->

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22:46:00

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* sqrt2 divided by sqrt 2 *sqrt 2

3 * 2^ 1/2 / 2

Good, but you want to say '3* sqrt2 multiplied by sqrt 2 *sqrt 2'.

You are multiplying, not dividing. Of course since the numerator and denominator are the same, you get the same result either way, so technically you're correct. But it would tend to complicate your thinking and lead to later errors to phrase this process in terms of division.

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22:46:11

** 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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23:25:41

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

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RESPONSE -->

sqrt2 *sqrt 7 -2 divided by sqrt 7 + 2 times sqrt 7 - 2

sqrt 14 - 2sqrt2 / 3

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23:26:07

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 -->

ok

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23:36:18

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 -->

multiply 3 by 1/6 =1/2

x^ 1/2 = sqrt x

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23:36:46

** 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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23:38:00

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

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RESPONSE -->

sqrt25^3 =125

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00:25:47

**** 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 -->

I could not find an explanation in the book that was adequate enough for me to solve this problem.

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00:26:03

06-19-2008 00:26:03

(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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NOTES ------->

This problem requires careful and step-by-step application of the laws of exponents, and you've got to be careful about the 'bookkeeping' aspect of the process, but you clearly know enough to do this.

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00:57:30

**** 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 -->

[ 81 -18x - [(x^2 /x - 2] ] /4

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01:05:02

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

=

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RESPONSE -->

[ (3 -x ) + 2x^2 - 3x ]/ 9

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01:13:34

**** 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 -->

a. 16.3 ft/sec

b. 32 ft/sec

c. 6.9 ft/sec

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01:14:38

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

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01:17:25

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 24 = cube rt 8 * cube rt 3

2 * cube rt 3

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01:18:09

** (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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01:26:44

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 -->

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01:33:30

** (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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01:39:34

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 -->

( x +4 ) ( x +4 ) = x^2+ 8x + 16

4 ( x^2 + 8x + 16 ) = 4x^2 + 32x + 64

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01:40:32

** 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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01:46:29

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

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RESPONSE -->

For me, this was very tough. The questions relating ot algebra in calcus I did not understand at all. At times, I don't think the problems were adequately explained in the r.8 section. After being out of college since 1978, the review has been most challenging.

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This is a challenging section. You're doing OK here. See my notes and let me know if you have questions.