Sarah Jones

course Mth 173

Hi :)

assignment #002

002. `query 2

College Algebra

05-31-2009

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13:09:32

query R.2.46 (was R.2.36) Evaluate for x = -2, and y = 3 the expression (2x - 3) / y and explan how you got your result.

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RESPONSE -->

(2(-2) - 3)/3

First we do inside the parentheses after we substitute the values of x and y into the expression

(-4 - 3)/3

Then we do the outer parentheses to have

-7/3

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13:09:52

** Starting with (2x-3)/y we substitute x=-2 and y=3 to get

(2*(-2) - 3)/3 =

(-4-3)/3=

-7/3. **

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13:15:48

query R.2. 55 (was R.2.45) Evaluate for x = 3 and y = -2: | |4x| - |5y| | and explan how you got your result.

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RESPONSE -->

First we substitute x for 3 and y for -2

| |4(3)| - |5(-2)|

| |12| - |-10| | The negative ten will be 10 (absolute value)

| 2 | = 2

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13:16:10

** Starting with | | 4x |- | 5y | | we substitute x=3 and y=-2 to get

| | 4*3 | - | 5*-2 | | =

| | 12 | - | -10 | | =

| 12-10 | =

| 2 | =

2. **

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13:18:11

query R.2.64 (was R.2.54) Explain what values, if any, must not be present in the domain of the expression (-9x^2 - x + 1) / (x^3 + x)

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RESPONSE -->

x cannot be equal to 0

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13:19:57

** The denominator of this expression cannot be zero, since division by zero is undefined.

Since x^3 + x factors into (x^2 + 1) ( x ) we see that x^3 + x = 0 only if x^2 + 1 = 0 or x = 0.

Since x^2 cannot be negative x^2 + 1 cannot be 0, so x = 0 is indeed the only value for which x^3 + x = 0. **

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13:29:44

query R.2.76 \ 73 (was R.4.6). What is -4^-2 and how did you use the laws of exponents to get your result?

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RESPONSE -->

-4^-2

First we use the reciprocal to make the exponent positive to have

(-1/4)^2 = -1/16 or -0.0625

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13:33:15

** order of operations implies exponentiation before multiplication; the - in front of the 4 is not part of the 4 but is an implicit multiplication by -1. Thus only 4 is raised to the -2 power.

-4^(-2) Since a^-b = 1 / (a^b), we have

4^-2 = 1 / (4)^2 = 1 / 16.

The - in front then gives us -4^(-2) = - ( 1/ 16) = -1/16.

If the intent was to take -4 to the -2 power the expression would have been written (-4)^(-2).**

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RESPONSE -->

I see where I put my parentheses around the whole numbers and I should have only put them around the denominator only.

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13:43:58

query Extra Problem. What is (3^-2 * 5^3) / (3^2 * 5) and how did you use the laws of exponents to get your result?

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RESPONSE -->

(125/9)/45

I made 3^-2 =1/3^2

(1/3^2 * 5^3) / (3^2 * 5)

(1/9 * 5^3) / (9 * 5)

(1/9 * 125) / (45)

(125/9) / (45)

= .3086419753

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13:49:53

** (3^(-2)*5^3)/(3^2*5). Grouping factors with like bases we have

3^(-2)/3^2 * 5^3 / 5. Using the fact that a^b / a^c = a^(b-c) we get

3^(-2 -2) * 5^(3-1), which gives us

3^-4 * 5^2. Using a^(-b) = 1 / a^b we get

(1/3^4) * 5^2. Simplifying we have

(1/81) * 25 = 25/81. **

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RESPONSE -->

I remember doing this problem this way in the book now that I see it written out again.

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14:16:28

query R.2.94. Express [ 5 x^-2 / (6 y^-2) ] ^ -3 with only positive exponents and explain how you used the laws of exponents to get your result.

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RESPONSE -->

((5x^-2)^-3)/(6y6-2)^-3

(5^-3(x^6))/(6^-3(y^6)

(216x^6)/(125y^6)

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14:17:25

[ 5 x^-2 / (6 y^-2) ] ^ -3 = (5 x^-2)^-3 / (6 y^-2)^-3, since (a/b)^c = a^c / b^c. This simplifies to

5^-3 (x^-2)^-3 / [ 6^-3 (y^-2)^-3 ] since (ab)^c = a^c b^c. Then since (a^b)^c = a^(bc) we have

5^-3 x^6 / [ 6^-3 y^6 ] . We rearrange this to get the result

6^3 x^6 / (5^3 y^6), since a^-b = 1 / a^b.

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14:24:48

query Extra Problem. Express (-8 x^3) ^ -2 with only positive exponents and explain how you used the laws of exponents to get your result.

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RESPONSE -->

(-8x^3)^-2

Distribute the exponent throughout the problem

-8^-2 *x^-6

Reciprocal of negative exponents

-1/8^2 * 1/x^6

-1/8^2 *x^6

-1/64*x^6

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14:27:25

** ERRONEOUS STUDENT SOLUTION: (-8x^3)^-2

-1/(-8^2 * x^3+2)

1/64x^5

INSTRUCTOR COMMENT:1/64x^5 means 1 / 64 * x^5 = x^5 / 64. This is not what you meant but it is the only correct interpretation of what you wrote.

Also it's not x^3 * x^2, which would be x^5, but (x^3)^2.

There are several ways to get the solution. Two ways are shown below. They make more sense if you write them out in standard notation.

ONE CORRECT SOLUTION: (-8x^3)^-2 =

(-8)^-2*(x^3)^-2 =

1 / (-8)^2 * 1 / (x^3)^2 =

1/64 * 1/x^6 =

1 / (64 x^5).

Alternatively

(-8 x^3)^-2 =

1 / [ (-8 x^3)^2] =

1 / [ (-8)^2 (x^3)^2 ] =

1 / ( 64 x^6 ). **

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RESPONSE -->

I see where I made the mistake of not keeping the negative sign with the 8. Because it was in parentheses it did belong just to the 8.

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14:33:34

query R.2.90 (was R.4.36). Express (x^-2 y) / (x y^2) with only positive exponents and explain how you used the laws of exponents to get your result.

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RESPONSE -->

= (1/(x)^2)(y)/(xy^2)

=x^-2-1*y1-2

=x^-3*y^-1

=1/x^3y

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14:33:51

** (1/x^2 * y) / (x * y^2)

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

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

= y / (x^3 y^2)

= 1 / (x^3 y).

Alternatively, or as a check, you could use exponents on term as follows:

(x^-2y)/(xy^2)

= x^-2 * y * x^-1 * y^-2

= x^(-2 - 1) * y^(1 - 2)

= x^-3 y^-1

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

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RESPONSE -->

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14:41:00

query Extra Problem. . Express 4 x^-2 (y z)^-1 / [ (-5)^2 x^4 y^2 z^-5 ] with only positive exponents and explain how you used the laws of exponents to get your result.

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RESPONSE -->

= 4/25* x^-2-4*y^-1-2*z^-1-5

=4/25*x^-6*y^-3*z^-6

=4/25(1/x^6y^3z^6)

=4/(25x^6y^3z^6)

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14:41:56

** Starting with

4x^-2(yz)^-1/ [ (-5)^2 x^4 y^2 z^-5] Squaring the -5 and using the fact that (yz)^-1 = y^1 * z^-1:

4x^-2 * y^-1 * z^-1/ [25 * x^4 * y^2 * z^-5} Grouping the numbers, and the x, the y and the z expression:

(4/25) * (x^-2/x^4) * (y^-1/y^2) * (z^-1/z^-5) Simplifying by the laws of exponents:

(4/25) * x^(-2-4) * y^(-1-2) * z^(-1+5) Simplifying further:

(4/25) * x^-6 * y^-3 * z^4 Writing with positive exponents:

4z^4/ (25x^6 * y^3 ) **

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RESPONSE -->

&#Your response did not agree with the given solution in all details, and you should therefore have addressed the discrepancy with a full self-critique, detailing the discrepancy and demonstrating exactly what you do and do not understand about the parts of the given solution on which your solution didn't agree, and if necessary asking specific questions (to which I will respond).

&#

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14:42:21

query R.2.122 (was R.4.72). Express 0.00421 in scientific notation.

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RESPONSE -->

4.21 X 10^-3

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14:42:33

** 0.00421 in scientific notation is 4.21*10^-3. This is expressed on many calculators as 4.21 E-4. **

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14:42:57

query R.2.128 (was R.4.78). Express 9.7 * 10^3 in decimal notation.

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RESPONSE -->

9700

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14:43:02

** 9.7*10^3 in decimal notation is 9.7 * 1000 = 9700 **

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14:46:29

query R.2.152 \ 150 (was R.2.78) If an unhealthy temperature is one for which | T - 98.6 | > 1.5, then how do you show that T = 97 and T = 100 are unhealthy?

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RESPONSE -->

|97-98.6|=|-1.6| = 1.6> 1.5

|100-98.6|=|1.4|=1.4<1.5

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14:46:50

** You can show that T=97 is unhealthy by substituting 97 for T to get | -1.6| > 1.5, equivalent to the true statement 1.6>1.5.

But you can't show that T=100 is unhealthy, when you sustitute for T then it becomes | 100 - 98.6 | > 1.5, or

| 1.4 | > 1.5, giving us

1.4>1.5, which is an untrue statement. **

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&#Good responses. See my notes and let me know if you have questions. &#