Questions assignment 2

course Mth 158

9-20-09 9:16 a.m.

If your solution to stated problem does not match the given solution, you should self-critique per instructions at http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.

Your solution, attempt at solution:

If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

002. `* 2

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

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Your solution:

(2x-3)/y when x = -2 and y = 3

(2(-2) – 3) / 3 substituting x,y you solve working in the order of operation

(-4-3) / 3

-7 / 3

confidence rating: 3

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Given Solution:

* * ** 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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Self-critique (if necessary):

There were no discrepancies

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Self-critique Rating: 3

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

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Your solution:

If x =3 and y =-2 then

The absolute value of 4x – the absolute value of 5y would be

4 * 3 – 5 * -2

12 – (-10)

12-10

2

Absolute value of the absolute value is 2

confidence rating: 3

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Given Solution:

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

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

| | 12 | - | -10 | | =

| 12-10 | =

| 2 | =

2. **

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

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Your solution:

(-9x^2 – x + 1) / (x^3 + x) In the denominator (x^3 + x) x cannot equal 0 because it is undefined

confidence rating: 3

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Given Solution:

* * ** 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 is, and only if, either 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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Self-critique (if necessary):

There were no discrepancies

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Self-critique Rating: 3

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Question:

* 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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Your solution:

-4^-2 Using the laws of exponents you would change the negative exponent by moving it

1/-4^2 = -1/16

confidence rating: 3

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Given Solution:

* * ** 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.

Starting with the expression -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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Self-critique (if necessary):

There were no discrepancies

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Self-critique Rating: 3

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Question:

* 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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Your solution:

(3^-2 * 5^3) / (3^2 * 5) Using the law of exponents, you would make the -2 positive by:

(1/3^2 * 5^3) / (3^2 * 5) Then solve using the order of operations

(1/9 * 125) / (9 * 5)

(125/9) / (45)

25/81

confidence rating: 3

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Given Solution:

Starting with (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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Self-critique (if necessary):

We solved the problem using different steps to obtain the same results. I don’t think either way is wrong, but the given solution gives more detailed steps than what I have. I seem to understand my way a little better, but worked the given solution on paper so I could see how it was solved.

Very good. Half the learning in an algebra-based math course occurs when you try to reconcile the given solution with your own.

Your solution was also fine.

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Self-critique Rating: 3

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Question:

* 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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Your solution:

[5x^-2 / (6y^-2)]^-3

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

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

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

From here I am not sure what to do next. I want to work it like this:

1/5^3 (x^6) / 1/6^3 (y^6)

But I don’t think that’s right.

confidence rating: 1

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Given Solution:

[ 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.

STUDENT QUESTION:

I do not see how you can take and seperate the problem down like this has it seems to just have reversed the problem

around in a different ordering and I do not see how this changed the exponets from being negative

Is there anyway you can explain this problem in a little more depth

INSTRUCTOR RESPONSE:

A fundamental law of exponents is that exponentiation distributes over multiplication, so that

(a * b) ^ c = a^c * b^c and

(a / b) ^ c = a^c / b^c

More specifically, if c = -3 then we have

( a * b ) ^ (-3) = a * (-3) * b^(-3) and

( a / b ) ^ (-3) = a ^ (-3) / b^(-3).

Now

a ^ (3) / b^(3) = 1 / a ^ (3) / (1 / b^(3)) and

1 / a ^ (3) / (1 / b^(3)) = 1 / a^3 * (b^3 / 1) = b^3 / a^3.

This principle applies to any string of multiplcations and division, so for example

( a * b / (c * d) ) ^ e = a^e * b^e / (c^e * d^e).

If e = -3 then we would have

( a * b / (c * d) ) ^ (-3) = a^(-3) * b^(-3) / (c^(-3) * d^(-3)).

Since the -3 power is the reciprocal of the 3 power this expression becomes

1/a^(3) * (1/b^(3)) / (1/c^(3) * (1/d^(3))), which is easily seen to be equal to

1 / (a^3 * b^3) / (1 / (c^3 * d^3) ).

Dividing by (1 / (c^3 * d^3) ) is the same as multiplying by (c^3 * d^3) / 1 so

1 / (a^3 * b^3) / (1 / (c^3 * d^3) ) = 1 / (a^3 * b^3) * (c^3 * d^3) = (c^3 * d^3) / (a^3 * b^3).

You should have written the above expressions, which are difficult to read in this notation, on paper, applying the order of operations. The expressions you wrote down should look like the ones below. Be sure you understand the translation from the 'typewriter notation' above to the standard notation depicted below, and be sure you know how to write each of the expressions depicted below in standard notation:

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Self-critique (if necessary):

I see now by the given solution, to make the -3 exponent a positive, you must moved it to the left and the one on the left was moved to the right making it the reciprocal.

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Self-critique Rating: I believe I have addressed all discrepancies between my solution and the given solution

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Question:

* 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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Your solution:

(-8x^3)^-2 you need to distribute here making it

-8^2 * x^-6 multiplying the x cubed by the -2 exponent when distributed

1/-8^2 * 1/x^6 here you make it divided by 1 to change the negative to positive exponent

1/-64 * 1/x^6

-64x^6

confidence rating:

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Given Solution:

* * ** 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^6).

Alternatively

(-8 x^3)^-2 =

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

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

1 / ( 64 x^6 ). **

I see where I made the mistake of keeping the answer a negative number, when you multiply a negative by a negative, you get a positive integer. The answer should have been 64x^6 a positive number.

* 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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Your solution:

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

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

This one has me confused. I am going to look at the given solution to see if I can understand how to solve.

confidence rating: 0

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Given Solution:

(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 positive and negative exponents, then in the last step express everything in terms of positive exponents, 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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Self-critique (if necessary):

I wrote it down on paper and I am still a little confused. I understand it down to the 3rd step and then I lose the meaning of the law of exponents.

Why does it change to:

(1/x^2 * y) multiplied by 1/xy^2 the multiplication throws me off.

(1/x^2 * y) means ( (1/x^2) * y, which is the same as (y / x^2).

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

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

Division by (x * y^2) is the same as multiplication by 1 / (x * y^2) .

So (y / x^2) / (x * y^2) means

(y / x^2) * (1 / (x * y^2)). Multiplying the numerators and denominators of these fractions we have

(y * 1) / (x^2 * x * y^2), which is

y / (x^3 * y^2). Dividing both numerator and denominator by y we have

1 / (x^2 * y).

Let me know if this doesn't help.

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Self-critique Rating: I will have to see a tutor on this one before I know that I have covered all the discrepancies.

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Question:

* 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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Your solution:

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

4/25 * x^-2/x^4 * y^-1/y^2 * z^-1/z^-5 using the law of exponents, you distribute to each variable

4/25 * x^-2-4) * y^-1-2) * z^(-1 – (-5)) here you just combine the exponents to make solving easier

4/25 * x^-6 * y^-3 * z^4 after subtracting the exponents, you move the negative exponents to the bottom of the equation leaving the positive z^4 on the top of the equation

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

confidence rating: 2

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Given Solution:

* * ** 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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Self-critique (if necessary):

This was a hard one for me. I wrote it on paper at least 3 times before I finally saw how to solve the last step. I had to keep going back to the book to find the laws of exponents before I understood it completely, and I’m still not sure I understand completely.

The only thing I see on the given solution that I did not do is put the ( ) around the 25x… on the last step. I’m not sure why it’s there other than understanding that it’s a given when you are dividing, but then wouldn’t you have to put it around the top of the equation also?

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Self-critique Rating: 3

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Question:

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

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Your solution:

4.21 * 10^-3

confidence rating: 3

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Given Solution:

* * ** 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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Self-critique (if necessary):

So far, and I’m sure it will get more complicated, scientific notation is one of the easiest things I’ve done.

There were no discrepancies in the given solution and mine.

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Self-critique Rating: 3

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Question:

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

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Your solution:

9,700

confidence rating: 3

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Given Solution:

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

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Self-critique (if necessary):

No discrepancies

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Self-critique Rating: 3

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Question:

* 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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Your solution:

Unhealthy temperature is:

| T – 98.6 | > 1.5

So, if your temperature T is:

| T = 97.00 – 98.6 = -1.6 | > 1.5 returns a True statement, so therefore you can show it is an unhealthy temperature

If you have T is = to:

| T = 100 – 98.6 = 1.4 | > 1.5 returns a False statement because 1.4 is not greater than 1.5

confidence rating: 3

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Given Solution:

* * ** 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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Self-critique (if necessary):

There are no discrepancies, although getting a return of a negative number sometimes gives me a hard time when I am solving for < or > and I’m not sure why.

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Self-critique Rating: 3

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