Ass8_Jaclyn Trout

course Math 158

[gwF}hrassignment #005

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.

005. `query 4

College Algebra

06-15-2008

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20:45:00

Query R.4.36 (was R.5.30). What is the single polynomial that is equal to 8 ( 4 x^3 - 3 x^2 - 1 ) - 6 ( 4 x^3 + 8 x - 2 )?

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RESPONSE -->

8 is the singly polynomial.

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20:45:25

** ERRONEOUS STUDENT SOLUTION: To make this problem into a single polynomial, you can group like terms together. (8-6)+ (4x^3-4x^3) + (-3x^2) + (8x) + (-1+2).

Then solve from what you just grouped...2 (-3x^2+8x+1).

INSTRUCTOR CORRECTION:

8 is multiplied by the first polynomial and 6 by the second. You can't isolate them like that.

Starting with

8 ( 4 x^3 - 3 x^2 - 1 ) - 6 ( 4 x^3 + 8 x - 2 ) use the Distributive Law to get

32 x^3 - 24 x^2 - 8 - 24 x^3 - 48 x + 12. Then add like terms to get

8x^3 - 24x^2 - 48x + 4 **

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RESPONSE -->

Ok.

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20:47:22

Query R.4.60 (was R.5.54). What is the product (-2x - 3) ( 3 - x)?

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RESPONSE -->

(-2x*3)(-2x-x)+(-3*3)(-3*-x)

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20:47:36

** Many students like to use FOIL but it's much better to use the Distributive Law, which will later be applied to longer and more complicated expressions where FOIL does not help a bit.

Starting with

(-2x - 3) ( 3 - x) apply the Distributive Law to get

-2x ( 3 - x) - 3 ( 3 - x). Then apply the Distributive Law again to get

-2x(3) - 2x(-x) - 3 * 3 - 3 ( -x) and simiplify to get

-6x + 2 x^2 - 9 + 3x. Add like terms to get

2 x^2 - 3 x - 9. **

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RESPONSE -->

Ok.

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20:50:55

Query R.4.66 (was R.5.60). What is the product (x - 1) ( x + 1) and how did you obtain your result using a special product formula?

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RESPONSE -->

(2x+x)+(-x-1)

3x+1

FOIL

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20:51:05

** Starting with

(x-1)(x+1) use the Distributive Law once to get

x ( x + 1) - 1 ( x+1) then use the Distributive Law again to get

x*x + x * 1 - 1 * x - 1 * 1. Simplify to get

x^2 +- x - x + - 1. Add like terms to get

x^2 - 1. **

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RESPONSE -->

Ok.

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20:52:04

Query R.4.84 (was R.5.78). What is (2x + 3y)^2 and how did you obtain your result using a special product formula?

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RESPONSE -->

2x^2+3y^2

Distributive Law

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20:52:40

** The Special Product is

(a + b)^2 = a^2 + 2 a b + b^2.

Letting a = 2x and b = 3y we get

(2x)^2 + 2 * (2x) * (3y) + (3y)^2, which we expand to get

4 x^2 + 12 x y + 9 y^2. **

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RESPONSE -->

Ok.

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20:57:43

Query R.4.105 \ 90 (was R.5.102). Explain why the degree of the product of two polynomials equals the sum of their degrees.

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RESPONSE -->

Because polynomials use exponents and they can multiply and add to be the same degree.

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20:57:53

** STUDENT ANSWER AND INSTRUCTOR COMMENTS: The degree of the product of two polynomials equals the sum of their degrees because you use the law of exponenents and the ditributive property.

INSTRUCOTR COMMENTS: Not bad.

A more detailed explanation:

The Distributive Law ensures that you will be multiplying the highest-power term in the first polynomial by the highest-power term in the second.

Since the degree of each polynomial is the highest power present, and since the product of two powers gives you an exponent equal to the sum of those powers, the highest power in the product will be the sum of the degrees of the two polynomials.

Since the highest power present in the product is the degree of the product, the degree of the product is the sum of the degrees of the polynomials. **

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RESPONSE -->

Ok.

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20:57:59

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

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RESPONSE -->

Nada.

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|^{טFD

assignment #005

005. `query 4

College Algebra

06-15-2008

ڙzJΪC

assignment #006

006. `query 6

College Algebra

06-15-2008

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21:03:36

R.5.22 (was R.6.18). What do you get when you factor 36 x^2 - 9 and how did you get your result?

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RESPONSE -->

-1.5

Factor out the sq.rt of 36, which equals 6. Divide 6x by 6 and 9 and that equals -9/6. Which could also be -1.5

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21:04:39

** 36x^2-9 is the difference of two squares. We write this as

(6x)^2-3^2 then get

(6x-3)(6x+3), using the special formula difference of two squars. **

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RESPONSE -->

Ok.

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21:05:42

R.5.32 \ 28 (was R.6.24 What do you get when you factor x^2 + 10 x + 1 and how did you get your result?

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RESPONSE -->

(x+5)+(x-5)+1

Difference of two squares.

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21:06:03

** STUDENT SOLUTION: x^2+10x+1 is prime because there are no integers whose product is 10 and sum is 1

INSTRUCTOR COMMENTS: The sum should be 10 and the product 1. I agree that there are no two integers with this property. Furthermore there are no two rational numbers with this property.

So you would never find the factors by inspection.

However that doesn't mean that there aren't two irrational numbers with the property. For example 10 and 1/10 come close, with product 1 and sum 10.1. The quadratic formula tells you in fact that the two numbers are ( -10 + sqrt( 10^4 - 4 * 1 * 1) ) / (2 * 1) and ( -10 - sqrt( 10^4 - 4 * 1 * 1) ) / (2 * 1) . Since 10^2 - 4 = 96 is positive, these are real numbers, both irrational. So the polynomial isn't prime. **

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RESPONSE -->

Ok.

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21:06:40

R.5.34 (was R.6.30). What do you get when you factor x^3 + 125 and how did you get your result?

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RESPONSE -->

It's prime.

Because there is no sq.rt

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21:06:51

** x^3+125 is the sum of two cubes, with 125 = 5^3. We know that a^3 + b^3 = (a+b) ( a^2 - 2 a b + b^2). So we write

x^3+5^3 = (x+5)(x^2-5x+25). **

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RESPONSE -->

Ok.

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21:14:31

R.5.46 (was R.6.42). What do you get when you factor x^2 - 17 x + 16 and how did you get your result?

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RESPONSE -->

(x-4)(x+4)

Difference of two squares.

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21:14:56

** x^2-17x+16 is of the form (x + a) ( x + b) = x^2 + (a + b) x + ab, with a+b = -17 and ab = 16.

If ab = 16 then we might have a = 1, b = 16, or a = 2, b = 8, or a = -2, b = -8, or a = 4, b = 4, or a = -1, b = -16, or a = -4, b = -4. These are the only possible integer factors of 16.

In order to get a + b = -17 we must have at least one negative factor. The only possibility that gives us a + b = -17 is a = -1, b = -16. So we conclude that

x^2 - 17 x + 16 = (x-16)(x-1). **

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RESPONSE -->

Ok.

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21:23:16

R.5.52 (was R.6.48). What do you get when you factor 3 x^2 - 3 x + 2 x - 2 and how did you get your result?

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RESPONSE -->

(x-1)(x+1.5)

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21:23:34

** This expression can be factored by grouping:

3x^2-3x+2x-2 =

(3x^2-3x)+(2x-2) =

3x(x-1)+2(x-1) =

(3x+2)(x-1). **

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RESPONSE -->

Ok.

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21:26:02

R.5.64 (was R.6.60). What do you get when you factor 3 x^2 - 10 x + 8 and how did you get your result?

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RESPONSE -->

Prime.

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21:26:28

** Possibilities are

(3x - 8) ( x - 1), (3x - 1) ( x - 8), (3x - 2) ( x - 4), (3x - 4) ( x - 2).

The possibility that gives us 3 x^2 - 10 x + 8 is (3x - 4) ( x - 2). **

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RESPONSE -->

Ok.

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21:31:35

R.5.82 (was R.6.78). What do you get when you factor 14 + 6 x - x^2 and how did you get your result?

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RESPONSE -->

(x+7)(x-7)

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21:31:48

** This expression factors, but not into binomtials with integer coefficients. We could list all the possibilities: (x + 7) ( -x + 2), (x + 2) ( -x + 7), (x + 14) ( -x + 1), (x + 1)(-x + 14), but none of these will give us the desired result.

For future reference:

You won't find the factors in the usual manner. The quadratic formula tells us that there are factors ( -6 + sqrt(6^2 - 4 * 14 * (-1) )) / (2 * -1) and ( -6 - sqrt(6^2 - 4 * 14 * (-1) ) ) / (2 * -1) .

Since sqrt(6^2 - 4 * 14 * (-1) ) = sqrt(36 + 56) = sqrt(92) is a real number these solutions are real numbers but again, as in a previous example, they aren't rational numbers and nobody could ever find them by inspection.

This is not something you're expected to do at this point. **

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RESPONSE -->

Ok.

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ش膑鷁ѲWܤ

assignment #006

006. `query 6

College Algebra

06-15-2008

}Oˉj}xi

assignment #007

007. `query 7

College Algebra

06-15-2008

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21:36:53

Query R.7.10 (was R.7.6). Show how you reduced (x^2 + 4 x + 4) / (x^4 - 16) to lowest terms.

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RESPONSE -->

(x+2)(x+2)/(x^2+8)(x^2-8)

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21:37:03

** We factor the denominator to get first (x^2-4)(x^2+4), then (x-2)(x+2)(x^2+4). The numerator factors as (x+2)^2. So the fraction is

(x+2)(x+2)/[(x-2)(x+2)(x^2+4)], which reduces to

(x+2)/[(x-2)(x^2+4)]. **

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RESPONSE -->

Ok.

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21:40:59

Query R.7.28 (was R.7.24). Show how you simplified[ ( x - 2) / (4x) ] / [ (x^2 - 4 x + 4) / (12 x) ].

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RESPONSE -->

{(x-2)/(4x)}/{(x^2-2)(x+2)

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21:41:16

** [ ( x - 2) / (4x) ] / [ (x^2 - 4 x + 4) / (12 x) ] =

(x-2) * / 4x * 12 x / (x^2 - 4x + 4) =

(x-2) * 12 x / [ 4x ( x^2 - 4x + 4) ] =

12 x (x-2) / [4x ( x-2) ( x-2) ] =

3/(x - 2) **

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RESPONSE -->

Ok.

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21:44:55

Query R.7.40 (was R.7.36). Show how you found and simplified the sum (2x - 5) / (3x + 2) + ( x + 4) / (3x + 2).

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RESPONSE -->

(2x-5+x+4)/(3x+2)

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21:45:20

** We have two like terms so we write

(2x-5)/(3x+2) + (x+4)/(3x+2) = [(2x-5)+(x+4)]/(3x+2). Simplifying the numerator we have

(3x-1)/(3x+2). **

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RESPONSE -->

Ok.

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21:47:08

Query R.7.52 (was R.7.48). Show how you found and simplified the expression(x - 1) / x^3 + x / (x^2 + 1).

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RESPONSE -->

[(x-1)+x]/[(x^2+x)(x+1)]

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21:47:21

** Starting with (x-1)/x^3 + x/(x^2+1) we multiply the first term by (x^2 + 1) / (x^2 + 1) and the second by x^3 / x^3 to get a common denominator:

[(x-1)/(x^3) * (x^2+1)/(x^2+1)]+[(x)/(x^2+1) * (x^3)/(x^3)], which simplifies to

(x-1)(x^2+1)/[ (x^3)(x^2+1)] + x^4/ [(x^3)(x^2+1)]. Since the denominator is common to both we combine numerators:

(x^3+x-x^2-1+x^4) / ) / [ (x^3)(x^2+1)] . We finally simplify to get

(x^4 +x^3 - x^2+x-1) / ) / [ (x^3)(x^2+1)] **

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RESPONSE -->

Ok.

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21:48:55

Query R.7.58 (was R.7.54). How did you find the LCM of x - 3, x^3 + 3x and x^3 - 9x, and what is your result?

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RESPONSE -->

(x^3+3)(x^3-3)

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21:49:14

** x-3, x^3+3x and x^3-9x factor into

x-3, x(x^2+3) and x(x^2-9) then into

(x-3) , x(x^2+3) , x(x-3)(x+3).

The factors x-3, x, x^2 + 3 and x + 3 'cover' all the factors of the three polynomials, and all are needed to do so. The LCM is therefore:

x(x-3)(x+3)(x^2+3) **

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RESPONSE -->

Ok.

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21:52:02

Query R.7.64 (was R.7.60). Show how you found and simplified the difference3x / (x-1) - (x - 4) / (x^2 - 2x + 1).

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RESPONSE -->

(x-1)-(x-4)/(x+1)(x-1)

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21:52:13

** Starting with 3x / (x-1) - (x-4) / (x^2 - 2x +1) we factor the denominator of the second term to obtain (x - 1)^2. To get a common denominator we multiply the first expression by (x-1) / (x-1) to get

3x(x-1)/(x-1)^2 - (x-4)/(x-1)^2, which gives us

(3x^2-3x-x-4) / (x-1)^2 = (3x^2 - 4x - 4) / (x-1)^2.

DRV**

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RESPONSE -->

Ok.

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21:52:39

QUESTION FROM STUDENT: On the practice test I'm having problems with problem #5 I don't know where to start or how to set it up. I'm probably missing something simple and will probably feel stupid by seeing the solution. Could you help with this problem.

A retailer is offering 35% off the purchase price of any pair of shoes during its annual charity sale. The sale price of the shoes pictured in the advertisement is $44.85. Find the original price of the shoes by solving the equation p-.35p = 44.85 for p.

INSTRUCTOR RESPONSE: It's very easy to get ahold of the wrong idea on a problem and then have trouble shaking it, or to just fail to look at it the right way. Nothing stupid about it, just human nature.

See if the following makes sense. If not let me know.

p - .35 p = 44.85. Since p - .35 p = 1 p - .35 p = (1 - .35) p = .65 p we have

.65 p = 44.85. Multiplying both sides by 1/.65 we get

p = 44.85 / .65 = etc. (you can do the division on your calculator); you'll get something near $67).

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RESPONSE -->

It makes sense.

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_NhwvEE

assignment #007

007. `query 7

College Algebra

06-15-2008

PޠXZɍԥ

assignment #008

008. `query 8

College Algebra

06-15-2008

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21:55:50

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

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RESPONSE -->

7.3

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21:56:08

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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21:58:37

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

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RESPONSE -->

y^2

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21:58:43

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 -->

Ok.

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21:59:13

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

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RESPONSE -->

-8.7

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21:59:19

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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RESPONSE -->

Ok.

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21:59:58

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 -->

58

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22:00:05

** (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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22:01:28

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

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RESPONSE -->

10

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22:01:35

(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.

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22:02:27

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 -->

2

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22:02:36

** 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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22:03:19

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

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RESPONSE -->

3

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22:03:25

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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22:05:01

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

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22:05:07

** 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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22:05:28

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

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RESPONSE -->

118

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22:05:34

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

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RESPONSE -->

Ok.

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22:06:28

**** 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 -->

(x^2) (y^2)

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22:06:35

(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.

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22:07:18

**** 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 -->

9x^2-9x^2

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22:07:24

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

=

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RESPONSE -->

Ok.

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22:07:52

**** 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 -->

8

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22:07:59

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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22:08:12

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

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RESPONSE -->

8

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22:08:16

** (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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22:09:32

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^2)/(1/3)

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22:09:36

** (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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22:10:24

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 -->

2x+1

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22:10:29

** 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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22:10:37

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

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RESPONSE -->

N/A

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KJÑص

assignment #008

008. `query 8

College Algebra

06-15-2008"

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