Assignment 2

course mth 158

Œ]มื—e‰ฺิŽ๐zิํ˜T|กมƒ๊ล€พฺassignment #002

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002. `query 2

College Algebra

06-10-2008

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12:14:26

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

(-4-3)/3

-7/3= -2.333....

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12:14:45

** 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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12:17:38

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 -->

l l 4(3)l - l5(-2) l l

l l 12 l - l -10l l

l 12 - 10 l

l 2 l = 2

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12:17:44

** 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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RESPONSE -->

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12:23:27

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 -->

0

9(0)^2-0+1)/(0^3 +0)=1/0

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12:24:10

** 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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12:25:26

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

1/-4^2

1/-16 or - 1/16

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12:25:33

** 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 -->

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12:58:11

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 -->

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

3^-4)*(5^2)

(1/3^4)*25

(1/81)*(25/1)=25/81=.308641975309

or

(1/(3^2)*125)/45

((1/9)*125)/45

13.8888..../45=.308641975309

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12:58:28

** (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 -->

you want to express your answers as fractions in lowest terms; decimal approximations aren't relevant in this section, though for some purposes they are acceptable

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13:03:58

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)/((6y^-2)^-3)

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

5x/6y

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13:10:37

[ 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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RESPONSE -->

I should of distributed the -3 to 5, x^-2, 6, and y^-2 individually and and then switched the top and bottom to eliminate the negatives.

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13:17:20

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 -->

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

(-8^-2)(x^-6)

-1/64)(1/x^6)

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13:19:58

** 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 -->

ok

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13:26:13

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 -->

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

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

(x/x^2)(y/y^2)

xy/(xy)^2

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13:43:08

** (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 -->

ok

(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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14:14:42

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 -->

4x^-2(yz)^-1/[(-5)^2 x^4y^2z^-5]

4x^-2y^-1Z^-1)/(-25x^4y^2z^-5)

(1/4 x^2 yz)(-25x^4y^2z^-5)

-100x^6y^3(1/z^4)

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14:26:33

** 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 -->

ok

(4x^-2y^-1z^-1)/[(-5)^2x^4y^2z^-5]

(4/25)(x^-2/x^4)(y^-1/y^2)(z^-1/z^-5)

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

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

4z^4/25x^6y^3

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14:27:00

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

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RESPONSE -->

4.21*10^3

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14:28:04

** 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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RESPONSE -->

I forgot to make it 10^-3

it should be this because the number is less than 1

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14:28:18

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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14:28:26

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

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14:31:07

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 -->

l 97-98.6 l>1.5

l -1.6 l > 1.5

1.6>1.5

this is true

l 100-98.6 l > 1.5

l 1.4 l > 1.5

1.4>1.5

this is not true

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14:31:21

** 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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Your work looks good. See my notes. Let me know if you have any questions. &#