Assignment 8

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.

......!!!!!!!!...................................

20:06:54 Extra question. What is the simplified form of sqrt( 4 ( x+4)^2 ) and how did you get this result?

......!!!!!!!!...................................

RESPONSE --> sqrt( 4 ( x+4)^2 ) = 2 [sqrt (x+2)^2] = 2(x + 2)

.................................................

......!!!!!!!!...................................

20:07:49 ** 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 | **

......!!!!!!!!...................................

RESPONSE --> I don't understand how you used the absolute value | | in this equation.

If x = 5, then sqrt(x^2) = x.

If x = -5, then sqrt(x^2) is still 5, but now sqrt(x^2) is not x but -x.

However in either case, sqrt(x^2) = | x |.

If x is negative, then -x is positive, so sqrt(x^2) = -x. If x is negative, then | x | = -x. So anytime x is negative, sqrt(x^2) = -x and sqrt(x^2) = | x |.

Of course when x is positive, sqrt(x^2) = x, which is the same thing as | x |.

.................................................

......!!!!!!!!...................................

20:08:30 Extra Question: What is the simplified form of (24)^(1/3) and how did you get this result?

......!!!!!!!!...................................

RESPONSE --> (24)^(1/3) = (8 * 3)^1/3 = 2 * 3^1/3

.................................................

......!!!!!!!!...................................

20:08:35 ** (24)^(1/3) = (8 * 3)^(1/3) = 8^(1/3) * 3^(1/3) = 2 * 3^(1/3) **

......!!!!!!!!...................................

RESPONSE --> ok

.................................................

......!!!!!!!!...................................

20:10:28 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?

......!!!!!!!!...................................

RESPONSE --> (x^2 y)^(1/3) * (125 x^3)^(1/3) / ( 8 x^3 y^4)^(1/3) = [(x^2/3 * y^1/3) * 5x] / (2 * y^4/3) I don't know where to go from this point.

.................................................

......!!!!!!!!...................................

20:12:03 ** (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) **

......!!!!!!!!...................................

RESPONSE --> I'm still a little confused about this problem. How do you get 5 (x^(5/3))?

x^(2/3) * x = x^(2/3) * x^1 = x^(2/3) * x^(3/3) = x^(2/3 + 3/3) = x^(5/3).

Write this out in standard notation if the 'typewriter' form is disconcerting.

.................................................

......!!!!!!!!...................................

20:13:38 Extra Question: What is the simplified form of 2 sqrt(12) - 3 sqrt(27) and how did you get this result?

......!!!!!!!!...................................

RESPONSE --> 2 sqrt(12) - 3 sqrt(27) = 2 sqrt(4 * 3) - 3 sqrt(9 * 3) = 2 * 2 sqrt(3) - 3 * 3 sqrt(3) = 4 sqrt(3) - 9 sqrt(3) = -5 sqrt(3)

.................................................

......!!!!!!!!...................................

20:13:44 ** 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. **

......!!!!!!!!...................................

RESPONSE --> ok

.................................................

......!!!!!!!!...................................

20:15:21 Extra Question: What is the simplified form of (2 sqrt(6) + 3) ( 3 sqrt(6)) and how did you get this result?

......!!!!!!!!...................................

RESPONSE --> (2 sqrt(6) + 3) ( 3 sqrt(6)) = 2 sqrt(6) * 3 sqrt(6) + 3 * 3 sqrt(6) = 6 sqrt(36) + 9 sqrt(6) = 6 * 6 + 9 sqrt(6) = 36 + 9 sqrt(6)

.................................................

......!!!!!!!!...................................

20:15:25 ** (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). **

......!!!!!!!!...................................

RESPONSE --> ok

.................................................

......!!!!!!!!...................................

20:16:34 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?

......!!!!!!!!...................................

RESPONSE --> 3 / sqrt(2) = 3 sqrt(2) / sqrt(2) * sqrt(2) = 3 sqrt(2) / 2

.................................................

......!!!!!!!!...................................

20:16:41 ** 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.

......!!!!!!!!...................................

RESPONSE --> ok

.................................................

......!!!!!!!!...................................

20:20:31 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?

......!!!!!!!!...................................

RESPONSE --> sqrt(3) / (sqrt(7) - sqrt(2)) = (sqrt(3) * (sqrt(7) + sqrt(2)) / (sqrt(7) - sqrt(2))(sqrt(7) + sqrt(2)) = (sqrt(3)*(sqrt(7) + sqrt(2)) / sqrt(49) + sqrt(14) - sqrt(14) - sqrt(4) = (sqrt(3)*(sqrt(7) + sqrt(2)) / 7 + 2 = (sqrt(3)*(sqrt(7) + sqrt(2)) / 9

.................................................

......!!!!!!!!...................................

20:21:49 ** 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.

......!!!!!!!!...................................

RESPONSE --> Why would you use sqrt(7) + 2 instead of the opposite of the denominator which would be sqrt(7) + sqrt(2)?

There is an editing error in that solution.

(sqrt(3)* (sqrt(7) + sqrt(2)))/ (sqrt(7) - sqrt(2))(sqrt(7) + sqrt(2)). Since (a-b)(a+b) = a^2 - b^2 the denominator is (sqrt(7)+sqrt(2) ) ( sqrt(7) - sqrt(2) ) = sqrt(7)^2 - sqrt(2)^2 = 7 - 2 = 5 so we have sqrt(3) (sqrt(7) + sqrt(2)) / 5.

Your solution should read

sqrt(3) / (sqrt(7) - sqrt(2)) = (sqrt(3) * (sqrt(7) + sqrt(2)) / [ (sqrt(7) - sqrt(2))(sqrt(7) + sqrt(2)) ] = (sqrt(3)*(sqrt(7) + sqrt(2)) / [ sqrt(49) + sqrt(14) - sqrt(14) - sqrt(4) ] = (sqrt(3)*(sqrt(7) + sqrt(2)) / [ 7 - 2 ] = (sqrt(3)*(sqrt(7) + sqrt(2)) / 5.

Note the grouping I added to your denominators.

.................................................

......!!!!!!!!...................................

20:23:01 Extra Question: What steps did you follow to simplify (-8)^(-5/3) and what is your result?

......!!!!!!!!...................................

RESPONSE --> (-8)^(-5/3) = (-8^1/3)^-5 = -2^-5 = 1 / -2^5 = - 1 / 32

.................................................

......!!!!!!!!...................................

20:23:06 ** (-8)^(-5/3) = [ (-8)^(1/3) ] ^-5. Since -8^(1/3) is -2 we get [-2]^-5 = 1 / (-2)^5 = -1/32. **

......!!!!!!!!...................................

RESPONSE --> ok

.................................................

......!!!!!!!!...................................

20:24:27 query R.8.64. What steps did you follow to simplify (8/27)^(-2/3) and what is your result?

......!!!!!!!!...................................

RESPONSE --> (8/27)^(-2/3) = 27^2/3 / 8^2/3 = (27^1/3)^2 / (8^1/3)^2 = 3^2 / 2^2 = 9/4

.................................................

......!!!!!!!!...................................

20:24:31 ** 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. **

......!!!!!!!!...................................

RESPONSE --> ok

.................................................

......!!!!!!!!...................................

20:25:07 Extra Question: What steps did you follow to simplify 6^(5/4) / 6^(1/4) and what is your result?

......!!!!!!!!...................................

RESPONSE --> 6^(5/4) / 6^(1/4) = 6^(5/4 - 1/4) = 6^1 = 6

.................................................

......!!!!!!!!...................................

20:25:15 ** 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. **

......!!!!!!!!...................................

RESPONSE --> ok

.................................................

......!!!!!!!!...................................

20:25:54 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?

......!!!!!!!!...................................

RESPONSE --> (x^3)^(1/6) = x^(3 * 1/6) = x^1/2

.................................................

......!!!!!!!!...................................

20:25:57 ** Express radicals as exponents and use the laws of exponents. (x^3)^(1/6) = x^(3 * 1/6) = x^(1/2). **

......!!!!!!!!...................................

RESPONSE --> ok

.................................................

......!!!!!!!!...................................

20:29:05 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?

......!!!!!!!!...................................

RESPONSE --> (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

.................................................

......!!!!!!!!...................................

20:29:21 ** (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. **

......!!!!!!!!...................................

RESPONSE --> ok

.................................................

......!!!!!!!!...................................

20:30:42 query R.8.96. Factor 8 x^(1/3) - 4 x^(-2/3), x <> 0.

......!!!!!!!!...................................

RESPONSE --> 8 x^(1/3) - 4 x^(-2/3) = 8x^(1/3 + 2/3) - 4x^(-2/3 + 2/3) = (8x - 4) / x^2/3

.................................................

......!!!!!!!!...................................

20:31:26 09-17-2006 20:31:26 ** 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). **

......!!!!!!!!...................................

NOTES -------> I would have goten the correct answer if I would have factored. I understand how you got that answer.

.......................................................!!!!!!!!...................................

20:31:27 ** 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). **

......!!!!!!!!...................................

RESPONSE -->

.................................................

......!!!!!!!!...................................

20:31:34 ** 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). **

......!!!!!!!!...................................

RESPONSE --> I would have gotten that answer if I would have factored. I understand what I did wrong.

.................................................

"

Good job. See my notes and let me know if you have questions.

Assignment 8

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.

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

If x = 5, then sqrt(x^2) = x.

If x = -5, then sqrt(x^2) is still 5, but now sqrt(x^2) is not x but -x.

However in either case, sqrt(x^2) = | x |.

If x is negative, then -x is positive, so sqrt(x^2) = -x.

If x is negative, then | x | = -x.

So anytime x is negative, sqrt(x^2) = -x and sqrt(x^2) = | x |.

Of course when x is positive, sqrt(x^2) = x, which is the same thing as | x |.

.................................................

......!!!!!!!!...................................

20:08:30

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

......!!!!!!!!...................................

RESPONSE -->

(24)^(1/3)

= (8 * 3)^1/3

= 2 * 3^1/3

.................................................

......!!!!!!!!...................................

20:08:35

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

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

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

2 * 3^(1/3) **

......!!!!!!!!...................................

RESPONSE -->

ok

.................................................

......!!!!!!!!...................................

20:10:28

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?

......!!!!!!!!...................................

RESPONSE -->

(x^2 y)^(1/3) * (125 x^3)^(1/3) / ( 8 x^3 y^4)^(1/3)

= [(x^2/3 * y^1/3) * 5x] / (2 * y^4/3)

I don't know where to go from this point.

.................................................

......!!!!!!!!...................................

20:12:03

** (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) **

......!!!!!!!!...................................

RESPONSE -->

I'm still a little confused about this problem. How do you get 5 (x^(5/3))?

x^(2/3) * x = x^(2/3) * x^1 = x^(2/3) * x^(3/3) = x^(2/3 + 3/3) = x^(5/3).

Write this out in standard notation if the 'typewriter' form is disconcerting.

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

There is an editing error in that solution.

(sqrt(3)* (sqrt(7) + sqrt(2)))/ (sqrt(7) - sqrt(2))(sqrt(7) + sqrt(2)). Since (a-b)(a+b) = a^2 - b^2 the denominator is (sqrt(7)+sqrt(2) ) ( sqrt(7) - sqrt(2) ) = sqrt(7)^2 - sqrt(2)^2 = 7 - 2 = 5 so we have

sqrt(3) (sqrt(7) + sqrt(2)) / 5.

Your solution should read

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

= (sqrt(3) * (sqrt(7) + sqrt(2)) / [ (sqrt(7) - sqrt(2))(sqrt(7) + sqrt(2)) ]

= (sqrt(3)*(sqrt(7) + sqrt(2)) / [ sqrt(49) + sqrt(14) - sqrt(14) - sqrt(4) ]

= (sqrt(3)*(sqrt(7) + sqrt(2)) / [ 7 - 2 ]

= (sqrt(3)*(sqrt(7) + sqrt(2)) / 5.

Note the grouping I added to your denominators.

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.......................................................!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

......!!!!!!!!...................................

......!!!!!!!!...................................

.................................................

Good job. See my notes and let me know if you have questions.