qa_12

#$&*

course Mth163

http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm

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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.

012.

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Question: `q001. Note that this assignment has 4 questions

If we know that y = k x^2, then if (x2/x1) = 7, what is (y2/y1)?

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Your solution:

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(y2/y1)=k(7^2)

(y2/y1)=k(49)

(1/49)(y2/y1)=k(49)(1/49)

y2/(49*y1)=k

Because if we can find k then we can figure out the y’s from there.

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confidence rating #$&*: 3

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Given Solution:

If y2 = k x2^2 and y1 = k x1^2, then y2 / y1 = (k x2^2) / ( k x1^2). Since k / k = 1 this is the same as

y2 / y1 = x2^2 / x1^2, which is the same as

y2 / y1 = (x2 / x1)^2.

In words this tells us if y to is proportional to the square of x, then the ratio of y2 to y1 is the same as the square of the ratio of x2 to x1.

Now if (x2 / x1) = 7, we see that y2 / y1 = (x2 / x1)^2 = 7^2 = 49.

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Self-critique (if necessary):

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Is the k in this case one as well or was that just an example???

Also did I do something wrong

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@&

The value of k doesn't matter.

y1 = k * x1 ^ 3

y2 = k * x2 ^ 3

What therefore is y2 / y1?

*@

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Self-critique rating:OK

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Question: `q002. If we know that y = k x^3, then if (x2/x1) = 7, what is (y2/y1)?

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Your solution:

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y2 / y1 = k(x2 / x1)^3 = 7^3

@&

The ratio y2 / y1 will not include k.

What happens if you divide the expression for y2 by the expression for y1?

*@

y2 / y1=k(343)

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confidence rating #$&*: 3

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Given Solution:

If y2 = k x2^3 and y1 = k x1^3, then y2 / y1 = (k x2^3) / ( k x1^3). Since k / k = 1 this is the same as

y2 / y1 = x2^3 / x1^3, which is the same as

y2 / y1 = (x2 / x1)^3.

In words this tells us if y to is proportional to the cube of x, then the ratio of y2 to y1 is the same as the cube of the ratio of x2 to x1.

Now if (x2 / x1) = 7, we see that y2 / y1 = (x2 / x1)^3 = 7^3 = 343.

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Self-critique (if necessary):

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Self-critique rating:OK

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Question: `q003. If we know that y = k x^-2, then if (x2/x1) = 64, what is (y2/y1)?

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Your solution:

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y2 / y1 = (x1 / x2)^2 = (1/64)^2 = 1/ 4096

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confidence rating #$&*: 3

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Given Solution:

If y2 = k x2^-2 and y1 = k x1^-2, then y2 / y1 = (k x2^-2) / ( k x1^-2). Since k / k = 1 this is the same as

y2 / y1 = x2^-2 / x1^-2, which is the same as

y2 / y1 = (x2 / x1)^-2, which is the same as

1 / (x2 / x1)^2, which gives us

(x1 / x2)^2.

So if y = k x^-2, then (y2 / y1) = (x1 / x2)^2.(

In words this tells us if y to is inversely proportional to the square of x, then the ratio of y2 to y1 is the same as the square of the ratio of x1 to x2 (note that this is a reciprocal ratio).

Now if (x2 / x1) = 64, we see that y2 / y1 = (x1 / x2)^2 = (1/64)^2 = 1/ 4096.

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Self-critique (if necessary):

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Self-critique rating:OK

If you understand the assignment and were able to solve the previously given problems from your worksheets, you should be able to complete most of the following problems quickly and easily. If you experience difficulty with some of these problems, you will be given notes and we will work to resolve difficulties.

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Question: `q004. If y = k / x and if y = 4 when x = 2, what is the value of y when x = 8?

What is the ratio of the new value of y to the original?

What is the ratio of the new value of x to the original?

If y = k / x and if (x2 / x1) = 3, then what is the value of (y2 / y1)?

In general how is the ratio y2 / y1 related to the ratio x2 / x1?

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Your solution:

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4=k/2

k=8

4=k/8

.k=32

@&

From the given information you have concluded that the value of k is 8.

That value does not change.

What is the proportionality between x and y, if k has the unchanging value 8?

What therefore is the value of y when x = 8?

*@

V stands for value

V2/V1

32/8=4

4 is the ratio

(y2 / y1)= k / ((x2 / x1) = 3)

(y2 / y1)= k / 3

The average of values is proportional to the square root of the average of x values

@&

y2 = k / x2, and

y1 = k / x1.

It does not follow that y2 / y1 = k / (x2 / x1).

What do you get when you divide the expression for y2 by the expression for y1?

*@

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confidence rating #$&*:

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Self-critique rating:N/A

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Self-critique (if necessary):

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Self-critique (if necessary):

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#*&!

@&

You have much of this but you aren't quite applying it correctly.

Look over all of my notes, and look more closely at the given solutions.

Then see if you can correct your work on the last two questions, on which I've included some leading questions.

&#Please see my notes and submit a copy of this document with revisions, comments and/or questions, and mark your insertions with &&&& (please mark each insertion at the beginning and at the end).

Be sure to include the entire document, including my notes.

&#

*@

qa_12

#$&*

course Mth163

Sorry if some of the words do not mark any sense I am using a word prosscer called pages to complete my work on my iPhone (auto correct may have put something silly in the place of my original words without me realizing, it does that a lot) June 30,2013 12:14

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.

013.

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Question: `q001. Note that this assignment has 14 questions

What does 2^5 mean?

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Your solution:

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2 raised to the fifth power (of 2)

2*2*2*2*2

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confidence rating #$&*:3

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Given Solution:

2^5 stands for 2 raised to the fifth power; i.e., 2^5 = 2*2*2*2*2.

The result of this calculation is 2^5 = 32.

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Self-critique (if necessary):

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The finial answer would be 32

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Self-critique rating:3

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Question: `q002. What does 2^3 * 2^5 mean? Is the result of power of 2? If so, what power of 2 is it?

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Your solution:

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When you have the same number with different powers you can just add the powers together if you are multiplying . So 2^3*2^5=2^8 =256

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confidence rating #$&*:3

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Given Solution:

2^3 * 2^5 means (2*2*2) * (2*2*2*2*2). This is the same as 2*2*2*2*2*2*2*2, or 2^8.

When we multiply this number out, we obtain 256.

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Self-critique (if necessary):

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Self-critique rating:OK

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Question: `q003. Why do we say that a^b * a^c = a^(b+c)?

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Your solution:

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When you multiply powers with the same large number it is exactly the same just to multiply the numbers out the long way except you are saving time this way.

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confidence rating #$&*:3

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Given Solution:

We saw in the preceding example that 2^3 * 2^5 stood for a product of three 2's, multiply by a product of five 2's. We saw also that the result was identical to a product of eight 2's. This was one instance of the general rule that when we multiply to different powers of the same number, the result is that number raised to the sum of the two powers.

One general way to state this rule is to let a stand for the number that is being raised to the different powers, and let b and c stand for those powers. Then we get the statement a^b * a^c = a^(b+c).

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Self-critique (if necessary):

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Self-critique rating:OK

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Question: `q004. What does (2^3)^5 mean?

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Your solution:

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This time you multiply the powers since you are trying to find a power of a power.

(2^3)^5 we could write 2*2*2=8, then 8^5=8*8*8*8*8=32,768

Or we could simply write 2^(3*5)=2^15=32,768

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confidence rating #$&*:3

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Given Solution:

Since 2^3 stands for 2*2*2, it follows that (2^3)^5 means (2^3)*(2^3)*(2^3)*(2^3)*(2^3) = (2*2*2)*(2*2*2)*(2*2*2)*(2*2*2)*(2*2*2) = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*2 = 2^15.

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Self-critique (if necessary):

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Self-critique rating:OK

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Question: `q005. Why do we say that (a^b)^c = a^(b*c)?

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Your solution:

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This time you multiply the powers since you are trying to find a power of a power.

@&

This is circular. You are identifying the operation as 'a power of a power', and that is good, but you're still essentially saying that the rule is so because it's so.

The rule is so because of the way numbers behave, and your explanation should be based on the fundamental properties of multiplication.

*@

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confidence rating #$&*:3

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Given Solution:

We saw in the last example how (2^3)^5 stands for the product of 5 quantities, each equal to the product of three 2's. We saw how this is equivalent to the product of fifteen 2's, and we saw how the fifteen was obtained by multiplying the exponents 3 and 5.

In the present question a^b stands for the quantity a multiplied by itself b times. (a^b)^c stands for the quantity a^b multiplied by itself c times, which is equivalent to multiplying a by itself b * c times. Thus we say that (a^b)^c = a^(b * c).

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Self-critique (if necessary):

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Self-critique rating:OK

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Question: `q006. According to the law a^b * a^c = a*(b+c), if we multiply 2^5 by 2^-2 what power of 2 should we get?

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Your solution:

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2^3=8

5+(-2)=3

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confidence rating #$&*:3

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Given Solution:

According to the law, 2^5 * 2^-2 = 2^(5 + -2) = 2^(5-2) = 2^3.

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Self-critique (if necessary):

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Self-critique rating:OK

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Question: `q007. Since as we saw in the preceding question 2^5 * 2^-2 = 2^3, what therefore must be the value of 2^-2?

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Your solution:

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When you take a negative power of a number you end up with the reciprocal of that number to that power.

2^-2= 1/4

Because 2*2=4 but it's neg - so you want to take the reciprocal of that number to make it positive which -4=1/4

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confidence rating #$&*:3

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Given Solution:

One way of seeing this is to note that 2^5 = 32 and 2^3 = 8, so we have 32 * 2^-2 = 8. Dividing both sides by 32 we get 2^-2 = 8 / 32 = 1/4.

We can learn something important if we keep the calculation in powers of 2. If 2^5 * 2^-2 = 2^3, then dividing both sides of the equation by 2^5 we obtain 2^-2 = 2^3/2^5, which is equal to 1/2^2.

This shows us why 2^-2 = 1/2^2.

QUESTIONABLE STUDENT RESPONSE: .25

INSTRUCTOR COMMENT

.25 is of course the value of 2^-2. However I'm not sure you've connected this with the fact that 2^5 * 2^-2 = 2^3, as was the intent of the question.

A key word in the given question is 'therefore', which asks you to connect your answer to the fact that 2^5 * 2^-2 = 2^3.

STUDENT QUESTION

It seems to me to be a lot more work to calculate 2^-2 in the process of solving 32 * 2^-2 = 8 but I understand why it can

also be done this way. I am just not sure I understand the need to see it this way. I guess I will understand “why” we need

to know this in future problems.

INSTRUCTOR RESPONSE

The distinction is between using a rule and understanding the reason for the rule.

The rule is that a^(-b) = 1 / a^b, so of course it's valid to say that 2^-2 = 1 / 2^2 = 1/4.

However just applying the rule doesn't give any insight into why the rule must be as it is.

The reason this has to be the rule is that if it isn't, then the calculation of this problem and others like it make no sense. If the other laws of exponents are to be consistent, a^(-b) must by 1 / a^b, as demonstrated by this example.

STUDENT QUESTION

I just don’t understand this part of our assignments. I went back to the CD to see if I could understand it further, but I

can’t seem to find the section. Is there a chapter in our book that would help me to better understand this?

INSTRUCTOR RESPONSE

We're trying to show why 2^(-n) = 1 / 2^n.

Think of it this way:

Suppose you have the expression, say, 2^3 / 2^7. We can look at this expression in two ways:

2^3 / 2^7, which by the laws of exponents must be 2^(3 - 7) = 2^(-4). Let's suppose we don't know what 2^(-4) means. We can

find out by looking at our original expression in another way:

2^3 / 2^7 = (2 * 2 * 2) / (2 * 2 * 2 * 2 * 2 * 2 * 2) = 1 / (2 * 2 * 2 * 2) = 1 / 2^4.

So now we have two ways of writing 2^3 / 2^7. One way is 2^(-4), the other way is 1 / 2^4.

We conclude that 2^(-4) = 1 / 2^4.

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Self-critique (if necessary):

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I should have connected it all together but over all I got the same answer anyway

32*2=64^-2=64*-64=-4,096=1/4,096

Okay now I'm confused why wouldn't we just put the finial answer over 1/4???

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Self-critique rating:1

@&

I'm not sure to what part of the above you're referring.

The given solution is as follows:

"One way of seeing this is to note that 2^5 = 32 and 2^3 = 8, so we have 32 * 2^-2 = 8. Dividing both sides by 32 we get 2^-2 = 8 / 32 = 1/4.

We can learn something important if we keep the calculation in powers of 2. If 2^5 * 2^-2 = 2^3, then dividing both sides of the equation by 2^5 we obtain 2^-2 = 2^3/2^5, which is equal to 1/2^2.

This shows us why 2^-2 = 1/2^2."

The last example explains in terms of a different calculation why 2^(-n) = 1 / 2^n.

*@

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Question: `q008. Why must we say that 2^-n = 1 / 2^n, where n stands for any counting number?

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Your solution:

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When you take a negative power of a number you end up with the reciprocal of that number to that power.

@&

That is what the statement says, but it's not why the statement is true.

*@

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confidence rating #$&*:3

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Given Solution:

This is because for any number m, we have 2^m * 2^-n = 2^( m + -n) = 2^(m-n), and we also have 2^m * (1 / 2^n) = 2^m / 2^n = 2^(m-n). So whether we multiply 2^m by 2^-n or by 1 / 2^n we get the same result. This shows that 2^-n and 1 / 2^n are the same.

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Self-critique (if necessary):

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Self-critique rating:OK

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Question: `q009. According to the law a^b * a^c = a^(b+c), if we multiply 2^3 by 2^-3 what power of 2 should we get?

Since 2^-3 = 1 / 2^3, what number must we get when we multiply 2^3 by 2^-3?

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Your solution:

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

2^0=1

@&

good

*@

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confidence rating #$&*:3

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Given Solution:

2^3 * 2^-3 = 2^(3 + -3) = 2^(3-3) = 2^0.

Since 2^-3 = 1 / 2^3 it follows that 2^3 * 2^-3 = 2^3 * ( 1 / 2^3) = 1.

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Self-critique (if necessary):

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The answer is one Because the reciprocal of the powers equal one

@&

The product of a number and its reciprocal is 1, by the definition of 'reciprocal'.

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Self-critique rating:3

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Question: `q010. Continuing the last question, what therefore should be the value of 2^0?

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Your solution:

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1

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confidence rating #$&*:3

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Given Solution:

Since 2^3 * 2^-3 = 2^0 and also 2^3 * 2^-3 = 1 we see that 2^0 must be 1.

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Self-critique (if necessary):

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Self-critique rating:OK

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Question: `q011. How do we solve the equation x^3 = 12?

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Your solution:

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. x^3=12

x^3(1/3)=12(1/3)

x=12/3=4

x=4

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confidence rating #$&*:2

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Given Solution:

We solve the equation by taking the 1/3 power of both sides:

(x^3)^(1/3) = 12^(1/3), then by the law (a^b)^c = a^(bc) we have

x^(3 * 1/3) = 12^(1/3), so that

x^1 = 12^(1/3) or just

x = 12^(1/3), which we can easily enough evaluate with a calculator. We obtain a result of approximately x = 2.29 .

STUDENT QUESTION

I see in the solution you raise each side to a power of 1/3. Is it also okay to just take the cube root of each side, since

that is equivalent?

INSTRUCTOR RESPONSE: It's easier to denote fractional exponents using the keyboard than to denote radicals, which is the reason fractional exponents are used in these solutions.

You should be able to do it either way, if requested.

However either way is fine, and your solution is good.

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Self-critique (if necessary):

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Raise it to the 1/3 power not multiply by the 1/3 power

12^(1/3)=1/36 would would it not be this???

@&

12^(1/3) is the number which, when raised to the power 3, is 12.

If you raise 36 to the power 3 you get something around 50 000. You don't get 12. (It would then follow that 50 000 raised to the 1/3 power would be about 36).

If you raise 1/36 to the power 3 you get about 1 / 50 000 or about .00002. It follows that .00002 ^ (1/3) is about 1/36.

*@

When I did it by cal to got 12^(1/3)=2.29

@&

That's right.

You should be able to estimate this.

2^3 = 8 so 8^(1/3) = 2.

12^(1/3) will be a bit more than 2.

3^3 = 27, so 27^(1/3) = 3.

12 is closer to 8 than to 27, so 12^(1/3) is closer to 2 than to 3.

*@

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Self-critique rating:2

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Question: `q012. How do we solve the equation x^(5/2) = 44?

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Your solution:

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(x^(5/2))(^2/5) =44^(2/5)

x^1=(44*2)/5

x=88/5

x=17.6

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confidence rating #$&*:3

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Given Solution:

If we take the 2/5 power of both sides we get

(x^(5/2))^(2/5) = 44^(2/5) or

x^(5/2 * 2/5) = 44^(2/5) or

x^1 = 44^(2/5) so that

x = 44^(2/5).

Evaluating by calculator you should obtain approximately x = 4.54.

STUDENT COMMENT

Okay, I see now why in question 11, you multiplied each side of the equation by 1/3 to get rid of the power of 3 attached to

x. In this problem it would be more difficult to take a 5/2 root of a number. It is much easier to just flip that fraction

and raise both sides to that power. (5/2) * (2/5) equals 1, so you wind up with x^1 on one side, which is just x and multiply

the other side by 2/5 to get the result of x.

INSTRUCTOR RESPONSE

You could equally well write this as the 5th root of x^2, or even as the square of the 5th root of x, but the fractional

notation is more compact, less cumbersome, in a text-based medium.

The fractional notation also makes more sense of the calculation, for the reasons you have noted.

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Self-critique (if necessary):

I keep wanting to multiply 2/5 by 44 and not take the power of but when I do raise the power I get 11^(2/5)*2^(4/5) or approximate 4.54

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Self-critique rating:3

@&

44^(2/5) is (44^2)^(1/5), which is about 2000 ^ (1/5).

4^5 is 1024, and 5^5 is 3125, both calculations you should be able to do in your head (something you should practice if, as is the case with nearly all high school students, you can't).

So 44^(2/5) is between 4 and 5.

Alternatively, 44^(2/5) is (44^(1/5))^2.

2^5 = 32 and 3^5 = 343, both calculations you should definitely be able to do in your head, so 44^(1/5) is between 2 and 3, much closer to 2 but a little bigger

So 44^(2/5) = (44^(1/5))^2 is a little bigger than 2^2 = 4.

Your calculation 11^(2/5) * 2^(4/5) is also valid, and is very good. It doesn't help much with estimating and understanding the value, but it shows good understanding of the laws.

*@

If you understand the assignment and were able to solve the previously given problems from your worksheets, you should be able to complete most of the following problems quickly and easily. If you experience difficulty with some of these problems, you will be given notes and we will work to resolve difficulties.

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Question: `q013. In terms of the meaning of 5^7, the meaning of 5^4 and the meaning of 5^11, explain why 5^7 * 5^4 = 5^11. We both know that the rule for multiplying these numbers tells us that 5^7 *5^4 = 5^(7 + 4). You can't explain by quoting this rule, or any rule; you need to explain in terms of the given meanings.

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Your solution:

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5^7=5*5*5*5*5*5*5=78125 5^4=5*5*5*5=625 78125*625=48828125 is the same as 5^11 = 48828125

5^(7+4)

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confidence rating #$&*:3

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Question: `q014. Explain in terms of the rule a^b * a^c = a^(b + c) why 7^-11 must be equal to 1 / (7^11).

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Your solution:

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When a^-n you are really diving by the power of your number so 1/(a^n)=a^-n

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confidence rating #$&*:3

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Question: Solve the equation 2 x * 5^(-3/5) = 9, giving an exact solution (which will be expressed in terms of rational numbers and powers of rational numbers; for example 17^(43/11) / 8 is expressed in this manner but its approximate value 8069.501481 is not exact).

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Your solution:

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2x(5^[-3/5])=9

2x(.381)=9

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This will not lead to an exact solution, since .381 is an approximation.

You need to find an expression for x that is exact. You can then simplify this expression if you wish.

How do you solve 2x(5^[-3/5])=9 for x, without any approximation? Your expression will be in terms of powers of 9 and/or 5.

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.762x=9

x=11.811

confidence rating #$&*:3

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Self-critique rating:N/A

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You're doing well with this, but you want to nail some of these laws down just a little tighter. So check out my notes.

The one problem I will ask you to revise is that last one.

Just submit a copy of that problem with your revised solution, and/or additional question, using a question form or a submit work form (either will be fine).

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